*6.3. Explicit Closed-Form Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[

To derive (explicit) closed-form lower bounds of the (nonexplicit) recursive lower bounds *B L λ*,*X*0,*n* for the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*) respectively closed-form upper bounds of the recursive upper bounds *B U λ*,*X*0,*n* for all parameters cases (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (PSP\PSP,1) × (R\{0, 1}), we combine part (b) of Theorem 1, Lemma 2, Lemma 3 together with appropriate parameters *p L <sup>λ</sup>* = *p L* (*β*A, *β*H, *α*A, *α*H, *λ*), *p U <sup>λ</sup>* = *p <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) <sup>≥</sup> <sup>0</sup> and *<sup>q</sup> L <sup>λ</sup>* = *q L* (*β*A, *β*H, *α*A, *α*H, *λ*), *q U <sup>λ</sup>* = *q <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) <sup>&</sup>gt; <sup>0</sup> satisfying (35). Notice that the representations of the lower and upper closed-form sequence-bounds depend on whether 0 < *q A <sup>λ</sup>* < *βλ*, 0 < *q A <sup>λ</sup>* = *β<sup>λ</sup>* or max{0, *βλ*} < *q A <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1} (*<sup>A</sup>* ∈ {*L*, *<sup>U</sup>*}).

Let us start with closed-form *lower* bounds for the case *λ* ∈]0, 1[; recall that the choice *p L <sup>λ</sup>* = *α λ* A *α* 1−*λ* H , *q L <sup>λ</sup>* = *β λ* A *β* 1−*λ* H led to the optimal recursive lower bounds *B L λ*,*X*0,*n* of the Hellinger integral (cf. Theorem 1(b) and Section 3.5). Correspondingly, we can derive

**Theorem 6.** *Let p<sup>L</sup> <sup>λ</sup>* = *α λ* A *α* 1−*λ* H *and q<sup>L</sup> <sup>λ</sup>* = *β λ* A *β* 1−*λ* H *. Then, the following assertions hold true:*

*(a) For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ <sup>P</sup>*SP,2* ∪ P*SP,3a* ∪ P*SP,3b* ∪ P*SP,3c* ×]0, 1[ *(for which particularly* 0 < *q L <sup>λ</sup>* < *βλ, β*<sup>A</sup> 6= *β*H*), all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *there holds*

*C* (*p L λ* ,*q L λ* ),*T <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≤ *C* (*p L λ* ,*q L λ* ),*L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≤ *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* < 1 , *where C*(*<sup>p</sup> L λ* ,*q L λ* ),*L λ*,*X*0,*n* := *C* (*p L λ* ,*q L λ* ),*T λ*,*X*0,*n* · exp ( *ζ* (*q L λ* ) *n* · *X*<sup>0</sup> + *p L λ q L λ* · *ϑ* (*q L λ* ) *n* ) (106) *with C*(*<sup>p</sup> L λ* ,*q L λ* ),*T λ*,*X*0,*n* :<sup>=</sup> exp ( *x* (*q L λ* ) 0 · " *X*<sup>0</sup> − *p L λ q L λ* · *d* (*q L λ* ),*T* 1 − *d* (*q L λ* ),*T* # · 1 − *d* (*q L λ* ),*T n* + *p L λ q L λ* · *β<sup>λ</sup>* + *x* (*q L λ* ) 0 − *α<sup>λ</sup>* ! · *n* ) , *and with ζ* (*q L λ* ) *n* := Γ (*q L λ* ) < · *d* (*q L λ* ),*T n*−<sup>1</sup> 1 − *d* (*q L λ* ),*T* · 1 − *d* (*q L λ* ),*T n* > 0 , (107) *ϑ* (*q L λ* ) *<sup>n</sup>* := Γ (*q L λ* ) < · 1 − *d* (*q L λ* ),*T n* 1 − *d* (*q L λ* ),*T* 2 · 1 − *d* (*q L λ* ),*T* 1 + *d* (*q L λ* ),*T n* 1 + *d* (*q L λ* ),*T* > 0 . (108)

*(b) For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP,4a* ∪ P*SP,4b*)×]0, 1[ *(for which particularly* 0 < *q L <sup>λ</sup>* = *βλ, β*<sup>A</sup> = *β*H*), all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *there holds*

$$\mathcal{C}^{(p\_{\lambda}^{\mathcal{L}},q\_{\lambda}^{\mathcal{L}}),\mathcal{L}}\_{\lambda,\mathcal{X}\_{0},\mathfrak{n}} := \mathcal{C}^{(p\_{\lambda}^{\mathcal{L}},q\_{\lambda}^{\mathcal{L}}),\mathcal{T}}\_{\lambda,\mathcal{X}\_{0},\mathfrak{n}} := \mathcal{B}^{\mathcal{L}}\_{\lambda,\mathcal{X}\_{0},\mathfrak{n}} = \exp\left\{ \left(p\_{\lambda}^{\mathcal{L}} - \mathfrak{a}\_{\lambda}\right) \cdot \mathfrak{n} \right\} < 1 \text{ .}$$

*(c) For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *one gets*

$$\begin{aligned} \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{C}^{(p\_{\lambda}^{L}, q\_{\lambda}^{L}), T}\_{\lambda, X\_{0}, n} \right) &= \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{C}^{(p\_{\lambda}^{L}, q\_{\lambda}^{L}), L}\_{\lambda, X\_{0}, n} \right) = \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{B}^{L}\_{\lambda, X\_{0}, n} \right) \\ &= \quad \frac{p\_{\lambda}^{L}}{q\_{\lambda}^{L}} \cdot \left( \mathcal{B}\_{\lambda} + \mathbf{x}\_{0}^{(q\_{\lambda}^{L})} \right) - a\_{\lambda} < 0 \,, \end{aligned}$$

*where in the case <sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*<sup>H</sup> *there holds q<sup>L</sup> <sup>λ</sup>* <sup>=</sup> *<sup>β</sup><sup>λ</sup> and x*(*<sup>q</sup> L λ* ) <sup>0</sup> = 0*.*

The proof will be provided in Appendix A.3.

In order to deduce closed-form *upper* bounds for the case *λ* ∈]0, 1[, we first recall from the Sections 3.6–3.13, that we have to employ suitable parameters *p U <sup>λ</sup>* = *p <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*), *<sup>q</sup> U <sup>λ</sup>* = *q <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) satisfying (35). Notice that we automatically obtain *<sup>p</sup> U <sup>λ</sup>* ≥ *p L <sup>λ</sup>* = *α λ* A *α* 1−*λ* <sup>H</sup> <sup>&</sup>gt; 0. Correspondingly, we obtain

**Theorem 7.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[*, all coefficients p U λ* , *q U <sup>λ</sup> which satisfy* (35) *for all x* ∈ N<sup>0</sup> *and additionally either* 0 < *q U <sup>λ</sup>* ≤ *β<sup>λ</sup> or β<sup>λ</sup>* < *q U <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1}*, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *the following assertions hold true:*

$$\mathcal{C}^{(p\_{\lambda}^{\mathrm{II}},q\_{\lambda}^{\mathrm{II}}),\mathcal{S}}\_{\lambda,X\_{0},\mathfrak{n}} \geq \mathcal{C}^{(p\_{\lambda}^{\mathrm{II}},q\_{\lambda}^{\mathrm{II}}),\mathrm{II}}\_{\lambda,X\_{0},\mathfrak{n}} \geq \widetilde{\mathcal{B}}^{(p\_{\lambda}^{\mathrm{II}},q\_{\lambda}^{\mathrm{II}})}\_{\lambda,X\_{0},\mathfrak{n}} \geq \mathcal{B}^{\mathrm{II}}\_{\lambda,X\_{0},\mathfrak{n}} \,\prime \qquad \text{where} \tag{109}$$

*(a) in the case* 0 < *q U <sup>λ</sup>* < *β<sup>λ</sup> one has*

*C* (*p U λ* ,*q U λ* ),*U λ*,*X*0,*n* := *C* (*p U λ* ,*q U λ* ),*S λ*,*X*0,*n* · exp ( − *ζ* (*q U λ* ) *n* · *X*<sup>0</sup> − *p U λ q U λ* · *ϑ* (*q U λ* ) *n* ) (110) *with C*(*<sup>p</sup> U λ* ,*q U λ* ),*S λ*,*X*0,*n* :<sup>=</sup> exp ( *x* (*q U λ* ) 0 · " *X*<sup>0</sup> − *p U λ q U λ* · *d* (*q U λ* ),*S* 1 − *d* (*q U λ* ),*S* # · 1 − *d* (*q U λ* ),*S n* + *p U λ q U λ* · *β<sup>λ</sup>* + *x* (*q U λ* ) 0 − *α<sup>λ</sup>* ! · *n* ) , *ζ* (*q U λ* ) *n* := Γ (*q U λ* ) < · *d* (*q U λ* ),*S n* − *d* (*q U λ* ),*T n d* (*q U λ* ),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q U λ* ),*T* − *d* (*q U λ* ),*S n*−<sup>1</sup> · 1 − *d* (*q U λ* ),*T n* 1 − *d* (*q U λ* ),*T* > 0 , (111) *ϑ* (*q U λ* ) *n* := Γ (*q U λ* ) < · *d* (*q U λ* ),*T* 1 − *d* (*q U λ* ),*T* · 1 − *d* (*q U λ* ),*S d* (*q U λ* ),*T n* 1 − *d* (*q U λ* ),*S d* (*q U λ* ),*T* − *d* (*q U λ* ),*S n* − *d* (*q U λ* ),*T n d* (*q U λ* ),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q U λ* ),*T* > 0 ; (112)

*furthermore, whenever p U λ* , *q U λ satisfy additionally* (47) *such parameters exist particularly in the setups* <sup>P</sup>*SP,2* ∪ P*SP,3a* ∪ P*SP,3b, cf. Sections 3.7–3.9 , then*

$$\mathbf{1} > \mathbf{C}^{(p\_{\lambda}^{\mathrm{II}}, \mathfrak{gl}\_{\lambda}^{\mathrm{II}}), \mathbf{S}}\_{\lambda, X\_{0}, \mathfrak{n}} \quad \text{and} \quad \widetilde{\mathcal{B}}^{(p\_{\lambda}^{\mathrm{II}}, \mathfrak{gl}\_{\lambda}^{\mathrm{II}})}\_{\lambda, X\_{0}, \mathfrak{n}} = \mathcal{B}^{\mathrm{II}}\_{\lambda, X\_{0}, \mathfrak{n}} \quad \forall \ n \in \mathbb{N};$$

*(b) in the case* 0 < *q U <sup>λ</sup>* = *β<sup>λ</sup> one has*

$$\mathcal{C}^{(p^{\mathcal{U}}\_{\lambda},q^{\mathcal{U}}\_{\lambda}),\mathcal{U}}\_{\lambda,X\_{0},\mathfrak{n}} := \mathcal{C}^{(p^{\mathcal{U}}\_{\lambda},q^{\mathcal{U}}\_{\lambda}),\mathcal{S}}\_{\lambda,X\_{0},\mathfrak{n}} := \hat{B}^{(p^{\mathcal{U}}\_{\lambda},q^{\mathcal{U}}\_{\lambda})}\_{\lambda,X\_{0},\mathfrak{n}} = \exp\left\{ \left(p^{\mathcal{U}}\_{\lambda} - \mathfrak{a}\_{\lambda}\right) \cdot \mathfrak{n} \right\};$$

*(c) in the case β<sup>λ</sup>* < *q U <sup>λ</sup>* <sup>&</sup>lt; min 1 , *e βλ*−1 *the formulas* (109) *and* (110) *remain valid, but with*

$$\overline{\zeta}\_{n}^{(q^{(l)}\_{\lambda})} := \quad \Gamma\_{\geqslant}^{(q^{(l)}\_{\lambda})} \cdot \left( d^{(q^{(l)}\_{\lambda}),S} \right)^{n-1} \cdot \left[ n - \frac{1 - \left( d^{(q^{(l)}\_{\lambda}),T} \right)^{n}}{1 - d^{(q^{(l)}\_{\lambda}),T}} \right] > 0 \,, \tag{113}$$

$$\begin{split} \overline{\Theta}\_{n}^{(q^{(l)}\_{\lambda})} &:= \quad \Gamma\_{\geqslant}^{(q^{(l)}\_{\lambda})} \cdot \left[ \frac{d^{(q^{(l)}\_{\lambda}),S} - d^{(q^{(l)}\_{\lambda}),T}}{\left(1 - d^{(q^{(l)}\_{\lambda}),S} \right)^{2} \left(1 - d^{(q^{(l)}\_{\lambda}),T} \right)} \cdot \left(1 - \left( d^{(q^{(l)}\_{\lambda}),S} \right)^{n} \right) \\ & \quad + \frac{d \left( q^{(q^{l}\_{\lambda}),T} \right) \left(1 - \left( d^{(q^{l}\_{\lambda}),S} d^{(q^{l}\_{\lambda}),T} \right)^{n} \right)}{\left(1 - d^{(q^{l}\_{\lambda}),T} \right) \left(1 - d^{(q^{l}\_{\lambda}),S} d^{(q^{l}\_{\lambda}),T} \right)} - \frac{\left( d^{(q^{l}\_{\lambda}),S} \right)^{n}}{1 - d^{(q^{l}\_{\lambda}),S}} \cdot n \right] > 0 \,, \tag{114} \end{split}$$

*(d) for all cases (a) to (c) one gets*

$$\begin{split} \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{C}^{(p\_{\lambda}^{\mathrm{II}}, q\_{\lambda}^{\mathrm{II}}), \mathcal{S}}\_{\lambda, X\_{0}, \mu} \right) &= \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{C}^{(p\_{\lambda}^{\mathrm{II}}, q\_{\lambda}^{\mathrm{II}}), \mathcal{U}}\_{\lambda, X\_{0}, \mu} \right) &= \lim\_{n \to \infty} \frac{1}{n} \log \left( \widetilde{\mathcal{B}}^{(p\_{\lambda}^{\mathrm{II}}, q\_{\lambda}^{\mathrm{II}})}\_{\lambda, X\_{0}, \mu} \right) \\ &= \frac{p\_{\lambda}^{\mathrm{II}}}{q\_{\lambda}^{\mathrm{II}}} \cdot \left( \beta\_{\lambda} + x\_{0}^{(q\_{\lambda}^{\mathrm{II}})} \right) - a\_{\lambda} \end{split}$$

*where in the case q<sup>U</sup> <sup>λ</sup>* <sup>=</sup> *<sup>β</sup><sup>λ</sup> there holds x*(*<sup>q</sup> U λ* ) <sup>0</sup> = 0*.*

This Theorem 7 will be proved in Appendix A.3. Notice that for an inadequate choice of *p U λ* , *q U λ* it may hold that *<sup>p</sup> U λ q U λ* (*β<sup>λ</sup>* + *x* (*q U λ* ) 0 ) − *α<sup>λ</sup>* > 0 in part (d) of Theorem 7.

*6.4. Explicit Closed-Form Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1])

For *λ* ∈ R\[0, 1], let us now construct closed-form *lower* bounds of the recursive lower bound components *B*e (*p L λ* ,*q L λ* ) *λ*,*X*0,*n* , for suitable parameters *p L <sup>λ</sup>* ≥ 0 and either 0 < *q L <sup>λ</sup>* ≤ *β<sup>λ</sup>* or max{0, *βλ*} < *q L <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1} satisfying (35).

**Theorem 8.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1]) *, all coefficients p L <sup>λ</sup>* ≥ 0, *q L <sup>λ</sup>* > 0 *which satisfy* (35) *for all x* ∈ N<sup>0</sup> *and either* 0 < *q L <sup>λ</sup>* ≤ *β<sup>λ</sup> or* max{0, *βλ*} < *q L <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1}*, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *the following assertions hold true:*

$$\mathcal{C}^{(p\_{\lambda}^{\mathcal{L}},\boldsymbol{\rho}\_{\lambda}^{\mathcal{L}}),\mathcal{T}}\_{\lambda,\boldsymbol{X\_{0},\boldsymbol{\mathfrak{u}}}} \leq \mathcal{C}^{(p\_{\lambda}^{\mathcal{L}},\boldsymbol{\rho}\_{\lambda}^{\mathcal{L}}),\mathcal{L}}\_{\lambda,\boldsymbol{X\_{0},\boldsymbol{\mathfrak{u}}}} \leq \widetilde{\mathcal{B}}^{(p\_{\lambda}^{\mathcal{L}},\boldsymbol{\rho}\_{\lambda}^{\mathcal{L}})}\_{\lambda,\boldsymbol{X\_{0},\boldsymbol{\mathfrak{u}}}} \leq \mathcal{B}^{\mathcal{L}}\_{\lambda,\boldsymbol{X\_{0},\boldsymbol{\mathfrak{u}}}} \,\prime \qquad\text{where}\tag{115}$$

*(a) in the case* 0 < *q L <sup>λ</sup>* < *β<sup>λ</sup> one has*

*C* (*p L λ* ,*q L λ* ),*L λ*,*X*0,*n* := *C* (*p L λ* ,*q L λ* ),*T λ*,*X*0,*n* · exp ( *ζ* (*q L λ* ) *n* · *X*<sup>0</sup> + *p L λ q L λ* · *ϑ* (*q L λ* ) *n* ) , (116) *with C*(*<sup>p</sup> L λ* ,*q L λ* ),*T λ*,*X*0,*n* :<sup>=</sup> exp ( *x* (*q L λ* ) 0 · " *X*<sup>0</sup> − *p L λ q L λ* · *d* (*q L λ* ),*T* 1 − *d* (*q L λ* ),*T* # · 1 − *d* (*q L λ* ),*T n* + *p L λ q L λ* · *β<sup>λ</sup>* + *x* (*q L λ* ) 0 − *α<sup>λ</sup>* ! · *n* ) *ζ* (*q L λ* ) *n* := Γ (*q L λ* ) < · *d* (*q L λ* ),*T n*−<sup>1</sup> 1 − *d* (*q L λ* ),*T* · 1 − *d* (*q L λ* ),*T n* > 0 , (117) *L n L L n* 

$$\mathfrak{G}\_{n}^{(q\_{\lambda}^{L})} := \Gamma\_{<}^{(q\_{\lambda}^{L})} \cdot \frac{1 - \left(d^{(q\_{\lambda}^{L}),T}\right)^{\circ}}{\left(1 - d^{(q\_{\lambda}^{L}),T}\right)^{2}} \cdot \left[1 - \frac{d^{(q\_{\lambda}^{L}),T} \left(1 + \left(d^{(q\_{\lambda}^{L}),T}\right)^{\circ}\right)}{1 + d^{(q\_{\lambda}^{L}),T}}\right] \\ > 0 \;;\tag{118}$$

*furthermore, whenever p L λ* , *q L λ satisfy additionally* (56) *such parameters exist particularly in the setups* P*SP,2* ∪ P*SP,3a* ∪ P*SP,3b, cf. Sections 3.17–3.19), then*

$$\mathbf{1} \prec \mathcal{C}^{(p\_{\lambda}^{\mathrm{L}}, q\_{\lambda}^{\mathrm{L}}), \mathcal{T}}\_{\lambda, \mathcal{X}\_{0}, \mathfrak{n}} \quad \text{and} \quad \tilde{\mathcal{B}}^{(p\_{\lambda}^{\mathrm{L}}, q\_{\lambda}^{\mathrm{L}})}\_{\lambda, \mathcal{X}\_{0}, \mathfrak{n}} = \mathcal{B}^{\mathrm{L}}\_{\lambda, \mathcal{X}\_{0}, \mathfrak{n}} \quad \forall \ n \in \mathbb{N};$$

*(b) in the case* 0 < *q L <sup>λ</sup>* = *β<sup>λ</sup> one has*

$$\mathcal{C}^{(p\_{\lambda}^{\mathrm{L}},q\_{\lambda}^{\mathrm{L}}),\mathsf{L}}\_{\lambda,X\_{0},\mathsf{n}} := \mathcal{C}^{(p\_{\lambda}^{\mathrm{L}},q\_{\lambda}^{\mathrm{L}}),\mathsf{T}}\_{\lambda,X\_{0},\mathsf{n}} = \widetilde{\mathcal{B}}^{(p\_{\lambda}^{\mathrm{L}},q\_{\lambda}^{\mathrm{L}})}\_{\lambda,X\_{0},\mathsf{n}} = \exp\left\{ \left(p\_{\lambda}^{\mathrm{L}} - \mathfrak{a}\_{\lambda}\right) \cdot \mathsf{n} \right\};$$

*(c) in the case* max{0 , *βλ*} < *q L <sup>λ</sup>* <sup>&</sup>lt; min 1 , *e βλ*−1 *the formulas* (115) *and* (116) *remain valid, but with*

$$\begin{split} \underline{\boldsymbol{\zeta}}\_{n}^{(q^{\boldsymbol{d}}\_{1})} &:= \boldsymbol{\Gamma}\_{>}^{(q^{\boldsymbol{d}}\_{1})} \cdot \frac{\left(d^{(q^{\boldsymbol{d}}\_{1}),T}\right)^{n} - \left(d^{(q^{\boldsymbol{d}}\_{1}),S}\right)^{2n}}{d^{(q^{\boldsymbol{d}}\_{1}),T} - \left(d^{(q^{\boldsymbol{d}}\_{1}),S}\right)^{2}} &> 0, \\ \underline{\boldsymbol{\varrho}}\_{n}^{(q^{\boldsymbol{d}}\_{1}),T} &:= \frac{\boldsymbol{\Gamma}\_{>}^{(q^{\boldsymbol{d}}\_{1})}}{d^{(q^{\boldsymbol{d}}\_{1}),T} - \left(d^{(q^{\boldsymbol{d}}\_{1}),S}\right)^{2}} \cdot \left[\frac{d^{(q^{\boldsymbol{d}}\_{1}),T} \cdot \left(1 - \left(d^{(q^{\boldsymbol{d}}\_{1}),T}\right)^{n}\right)}{1 - d^{(q^{\boldsymbol{d}}\_{1}),T}} - \frac{\left(d^{(q^{\boldsymbol{d}}\_{1}),S}\right)^{2} \cdot \left(1 - \left(d^{(q^{\boldsymbol{d}}\_{1}),S}\right)^{2n}\right)}{1 - \left(d^{(q^{\boldsymbol{d}}\_{1}),S}\right)^{2}}\right] &> 0 \; : \end{split} \tag{120}$$

*(d) for all cases (a) to (c) one gets*

$$\begin{split} \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathbb{C}^{(p\_{\lambda}^{\mathrm{L}}, q\_{\lambda}^{\mathrm{L}}), T}\_{\lambda, X\_{0, n}} \right) &= \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathbb{C}^{(p\_{\lambda}^{\mathrm{L}}, q\_{\lambda}^{\mathrm{L}}), L}\_{\lambda, X\_{0, n}} \right) = \lim\_{n \to \infty} \frac{1}{n} \log \left( \widehat{\mathcal{B}}\_{\lambda, X\_{0, n}}^{(p\_{\lambda}^{\mathrm{L}}, q\_{\lambda}^{\mathrm{L}})} \right) \\ &= \quad \frac{p\_{\lambda}^{\mathrm{L}}}{q\_{\lambda}^{\mathrm{L}}} \cdot \left( \beta\_{\lambda} + x\_{0}^{(q\_{\lambda}^{\mathrm{L}})} \right) - a\_{\lambda} \end{split}$$

*where in the case q<sup>L</sup> <sup>λ</sup>* <sup>=</sup> *<sup>β</sup><sup>λ</sup> there holds x*(*<sup>q</sup> L λ* ) <sup>0</sup> = 0*.*

For the proof of Theorem 8, see Appendix A.3. Notice that for an inadequate choice of *p L λ* , *q L λ* it may hold that *<sup>p</sup> L λ q L λ* (*β<sup>λ</sup>* + *x* (*q U λ* ) 0 ) − *α<sup>λ</sup>* < 0 in the last assertion of Theorem 8.

To derive closed-form *upper* bounds of the recursive upper bounds *B U λ*,*X*0,*n* of the Hellinger integral in the case *λ* ∈ R\[0, 1] , let us first recall from Section 3.24 that we have to use the parameters *p U <sup>λ</sup>* = *α λ* A *α* 1−*λ* <sup>H</sup> <sup>&</sup>gt; <sup>0</sup> and *<sup>q</sup> U <sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; 0. Furthermore, in the case *<sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*<sup>H</sup> we obtain from Lemma <sup>1</sup> (setting *q<sup>λ</sup>* = *q U λ* ) the assertion that max{0, *βλ*} < *q U <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1} iff *<sup>λ</sup>* <sup>∈</sup>]*λ*−, *<sup>λ</sup>*+[ \ [0, 1] implying that the sequence *a* (*q U λ* ) *n n*∈N converges . In the case *β*<sup>A</sup> = *β*<sup>H</sup> on gets *q U <sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>=</sup> *<sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*<sup>H</sup> <sup>=</sup> *<sup>β</sup><sup>λ</sup>* and therefore (cf. (P2)) *a* (*q U λ* ) *<sup>n</sup>* = 0 for all *n* ∈ N and for all *λ* ∈ R\[0, 1]. Correspondingly, we deduce

**Theorem 9.** *Let p<sup>U</sup> <sup>λ</sup>* = *α λ* A *α* 1−*λ* H *and q<sup>U</sup> <sup>λ</sup>* = *β λ* A *β* 1−*λ* H *. Then, the following assertions hold true:*

*U*

*U*

*(a) For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP,2* ∪ P*SP,3a* ∪ P*SP,3b* ∪ P*SP,3c*) × ( ]*λ*−, *λ*+[ \[0, 1] ) *(in particular for β*<sup>A</sup> 6= *β*H*), all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *there holds*

$$\infty > \mathcal{C}^{(p\_{\lambda}^{\mathcal{U}}, q\_{\lambda}^{\mathcal{U}}), \mathcal{S}}\_{\lambda, X\_{0}, \mathfrak{n}} \ge \mathcal{C}^{(p\_{\lambda}^{\mathcal{U}}, q\_{\lambda}^{\mathcal{U}}), \mathcal{U}}\_{\lambda, X\_{0}, \mathfrak{n}} \ge \mathcal{B}^{\mathcal{U}}\_{\lambda, X\_{0}, \mathfrak{n}} > 1 \,\mathsf{A}$$

*where C*(*<sup>p</sup>*

*U*

*U*

*λ* ,*q λ* ),*U λ*,*X*0,*n* := *C* (*p λ* ,*q λ* ),*S λ*,*X*0,*n* · exp ( − *ζ* (*q λ* ) *n* · *X*<sup>0</sup> − *λ q U λ* · *ϑ* (*q λ* ) *n* (121) *with C*(*<sup>p</sup> U λ* ,*q U λ* ),*S λ*,*X*0,*n* :<sup>=</sup> exp ( *x* (*q U λ* ) 0 · " *X*<sup>0</sup> − *p U λ q U λ* · *d* (*q U λ* ),*T* 1 − *d* (*q U λ* ),*T* # · 1 − *d* (*q U λ* ),*T n* + *p U λ q U λ* · *β<sup>λ</sup>* + *x* (*q U λ* ) 0 − *α<sup>λ</sup>* ! · *n* ) , *ζ* (*q U λ* ) *n* := Γ (*q U λ* ) > · *d* (*q U λ* ),*S n*−<sup>1</sup> · *n* − 1 − *d* (*q U λ* ),*T n* 1 − *d* (*q U λ* ),*T* > 0 , (122) *ϑ* (*q U λ* ) *n* := Γ (*q U λ* ) > · " *d* (*q U λ* ),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q U λ* ),*T* 1 − *d* (*q U λ* ),*S* <sup>2</sup> 1 − *d* (*q U λ* ),*T* · 1 − *d* (*q U λ* ),*S n* + *d* (*q U λ* ),*T* 1 − *d* (*q U λ* ),*S d* (*q U λ* ),*T n* 1 − *d* (*q U λ* ),*T* <sup>1</sup> <sup>−</sup> *<sup>d</sup>* (*q U λ* ),*S d* (*q U λ* ),*T* − *d* (*q U λ* ),*S n* 1 − *d* (*q U λ* ),*S* · *n* # > 0 . (123)

*U*

*p U*

*U*

)

*(b) For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP,4a* ∪ P*SP,4b*) × ( R\[0, 1] ) *(for which particularly* 0 < *q U <sup>λ</sup>* = *βλ, β*<sup>A</sup> = *β*H*), all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *there holds*

$$\mathcal{C}^{(p^{\mathrm{II}}\_{\lambda}\eta^{\mathrm{II}}\_{\lambda}),\mathsf{U}}\_{\lambda,X\_{0},\mathsf{u}} := \mathcal{C}^{(p^{\mathrm{II}}\_{\lambda}\eta^{\mathrm{II}}\_{\lambda}),\mathsf{S}}\_{\lambda,X\_{0},\mathsf{u}} := \mathcal{B}^{\mathrm{II}}\_{\lambda,X\_{0},\mathsf{u}} = \exp\left\{ \left(p^{\mathrm{II}}\_{\lambda} - \mathsf{a}\_{\lambda}\right) \cdot \mathsf{n} \right\} > 1 \ .$$

*(c) For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × ( ]*λ*−, *λ*+[ \[0, 1] ) *and all initial population sizes X*<sup>0</sup> ∈ N *one gets*

$$\begin{split} \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathbb{C}^{(p\_{\lambda}^{\mathrm{II}}, \rho\_{\lambda}^{\mathrm{II}}), S}\_{\lambda, X\_{0}, n} \right) &= \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathbb{C}^{(p\_{\lambda}^{\mathrm{II}}, \rho\_{\lambda}^{\mathrm{II}}), \mathrm{II}}\_{\lambda, X\_{0}, n} \right) = \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{B}\_{\lambda, X\_{0}, n}^{\mathrm{II}} \right) \\ &= \frac{p\_{\lambda}^{\mathrm{II}}}{q\_{\lambda}^{\mathrm{II}}} \cdot \left( \mathcal{B}\_{\lambda} + \mathbf{x}\_{0}^{(q\_{\lambda}^{\mathrm{II}})} \right) - a\_{\lambda} > 0 \,, \end{split}$$

*where in the case <sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*<sup>H</sup> *there holds q<sup>U</sup> <sup>λ</sup>* <sup>=</sup> *<sup>β</sup><sup>λ</sup> and x*(*<sup>q</sup> U λ* ) <sup>0</sup> = 0*.*

A proof of Theorem 9 is provided in Appendix A.3.

**Remark 7.** *Substituting a* (*q*) *<sup>n</sup> by a* (*q*),*T <sup>n</sup> resp. a* (*q*),*S <sup>n</sup> (cf.* (78) *resp.* (79)*) in B*e (*p*,*q*) *λ*,*X*0,*n from* (42) *leads to the "rudimentary" closed-form bounds C* (*p*,*q*),*T λ*,*X*0,*n resp. C* (*p*,*q*),*S λ*,*X*0,*n , whereas substituting a* (*q*) *<sup>n</sup> by a* (*q*) *<sup>n</sup> resp. a* (*q*) *<sup>n</sup> (cf.* (92) *resp.* (94)*) in B*e (*p*,*q*) *λ*,*X*0,*n from* (42) *leads to the "improved" closed-form bounds C* (*p*,*q*),*L λ*,*X*0,*n resp. C* (*p*,*q*),*U λ*,*X*0,*n in all the Theorems 5–9.*

#### *6.5. Totally Explicit Closed-Form Bounds*

The above-mentioned results give closed-form lower bounds *C* (*p*,*q*),*L λ*,*X*0,*n* , *C* (*p*,*q*),*T λ*,*X*0,*n* resp. closed-form upper bounds *C* (*p*,*q*),*U λ*,*X*0,*n* , *C* (*p*,*q*),*S λ*,*X*0,*n* of the Hellinger integrals *Hλ*(*P*A,*n*||*P*H,*n*) for case-dependent choices of *p*, *q*. However, these bounds still involve the fixed point *x* (*q*) <sup>0</sup> which in general has to be calculated implicitly. In order to get "totally" explicit but "slightly" less tight closed-form bounds of *Hλ*(*P*A,*n*||*P*H,*n*), one can proceed as follows:


For instance, one can use the following choices which will be also employed as an auxiliary tool for the diffusion-limit-concerning proof of Lemma A6 in Appendix A.4:

$$\begin{split} \underline{\boldsymbol{\omega}}\_{0}^{(q)} &:= \begin{cases} \begin{array}{ll} q^{-1} \cdot \boldsymbol{\varepsilon} \cdot \boldsymbol{\varepsilon}\_{\boldsymbol{\Delta}}^{(q)} \cdot \left[ (1-q) - \sqrt{(1-q)^{2} - 2 \cdot q \cdot \boldsymbol{\varepsilon}\_{\boldsymbol{\Delta}}^{(q)}} \cdot (q-\beta\lambda) \right], & \text{if} \quad q \in ]0, \beta\lambda\_{\boldsymbol{\Delta}} \| \boldsymbol{\varepsilon}\_{\boldsymbol{\Delta}} \end{array} \\\\ \begin{array}{ll} q^{-1} \cdot \left[ (1-q) - \sqrt{(1-q)^{2} - 2 \cdot q \cdot (q-\beta\lambda)} \right], & \text{if} \quad \max\{0, \beta\_{\boldsymbol{\Delta}}\} < q < \min\{1, \boldsymbol{\varepsilon}^{\beta\_{\boldsymbol{\Delta}}-1} \}, \\\\ \text{where} \quad \underline{\boldsymbol{\varepsilon}}\_{0}^{(q)} := \begin{cases} \begin{array}{ll} \max\left\{ -\beta\_{\lambda} \cdot \left( \frac{q-\beta\lambda}{1-q} \right) \right\}, & \text{if } q \in ]0, 1 \mid \\ -\beta\_{\lambda'} & \text{if } q \ge 1, \end{cases} \\\\ \end{array} \\\\ \begin{array}{ll} \begin{array}{ll} q^{-1} \cdot \left[ (1-q) - \sqrt{(1-q)^{2} - 2 \cdot q \cdot (q-\beta\_{\lambda})} \right], & \text{if } q \in ]0, \beta\_{\lambda} \mid \end{array} \end{cases} \end{split} \tag{125}$$

$$
\overline{\boldsymbol{x}}\_{0}^{(q)} := \begin{cases}
\boldsymbol{q} & \text{if } \left[ (1-q) - \sqrt{(1-q)^{2} - 2 \cdot q} \cdot (\boldsymbol{q} - \boldsymbol{\beta}\_{\lambda}) \right], & \text{if } \quad \boldsymbol{q} \in [0, \boldsymbol{\beta}\_{\lambda}], \\\\
(1-q) - \sqrt{(1-q)^{2} - 2 \cdot (q - \boldsymbol{\beta}\_{\lambda})}, & \text{if } \quad \max\{0, \boldsymbol{\beta}\_{\lambda}\} < q < \min\{1, \boldsymbol{\epsilon}\_{\lambda}^{\boldsymbol{\beta}\_{\lambda} - 1}\} \\\\
\overline{\boldsymbol{x}}\_{0}^{(q)} := -\log(q) & \text{if } \quad \max\{0, \boldsymbol{\beta}\_{\lambda}\} < q < \min\{1, \boldsymbol{\epsilon}\_{\lambda}^{\boldsymbol{\beta}\_{\lambda} - 1}\} \\
& \text{and } (1-q)^{2} - 2 \cdot q \cdot (q - \boldsymbol{\beta}\_{\lambda}) < 0.
\end{cases} \tag{126}
$$

Behind this choice "lies" the idea that–in contrast to the solution *x* (*q*) 0 of *ξ* (*q*) *λ* (*x*) :<sup>=</sup> *qe<sup>x</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>=</sup> *x*–the point *x* (*q*) 0 is a solution of (the obviously explicitly solvable) *Q* (*q*) *λ* (*x*) := *a* (*q*) *λ x* <sup>2</sup> + *b* (*q*) *λ x* + *c* (*q*) *<sup>λ</sup>* = *x* in both cases 0 < *q* < *β<sup>λ</sup>* and max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1}, whereas the point *<sup>x</sup>* (*q*) 0 is a solution of *Q* (*q*) *λ* (*x*) := *a* (*q*) *λ x* <sup>2</sup> + *b* (*q*) *λ x* + *c* (*q*) *<sup>λ</sup>* = *x* in the case 0 < *q* < *β<sup>λ</sup>* and in the case max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1} together with (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*) <sup>2</sup> <sup>−</sup> <sup>2</sup> · *<sup>q</sup>* · (*<sup>q</sup>* <sup>−</sup> *<sup>β</sup>λ*) <sup>≥</sup> 0. Thereby, *<sup>Q</sup>* (*q*) *λ* (·) and *Q* (*q*) *λ* (·) are the lower resp. upper quadratic approximates of *ξ* (*q*) *λ* (·) satisfying the following constraints:

• for *q* ∈]0, *βλ*[ (mostly but not only for *λ* ∈]0, 1[) (lower bound):

$$\underline{Q}\_{\lambda}^{(q)}(0) = \mathfrak{f}\_{\lambda}^{(q)}(0) = q - \mathfrak{f}\_{\lambda'} \qquad \underline{Q}\_{\lambda}^{(q)}(0) = \mathfrak{f}\_{\lambda}^{(q)} \, ^\prime(0) = q \, \qquad \underline{Q}\_{\lambda}^{(q)} \, ^\prime(\mathbf{x}) = \mathfrak{f}\_{\lambda}^{(q)} \, ^\prime(\mathbf{y}) = qe^{\mathbf{y}} \, \quad \mathbf{x} \in \mathbb{R}\_{+} $$

for some explicitly known approximate *y* < *x* (*q*) 0 leading to the (tighter) explicit lower approximate *x* (*q*) <sup>0</sup> ∈]*y*, *x* (*q*) 0 [ ; here, we choose

$$y := \underline{\mathfrak{x}}\_0^{(q)} := \begin{cases} \max\left\{-\beta\_\lambda \, , \, \frac{q - \beta\_\lambda}{1 - q}\right\}, & \text{if } q < 1, \\\ -\beta\_\lambda \, , & \text{if } q \ge 1; \end{cases}$$

• for *q* ∈]0, *βλ*[ (mostly but not only for *λ* ∈]0, 1[) (upper bound):

$$\overline{\mathcal{Q}}\_{\lambda}^{(q)}(0) = \mathfrak{f}\_{\lambda}^{(q)}(0) = q - \beta\_{\lambda \prime} \qquad \overline{\mathcal{Q}}\_{\lambda}^{(q) \prime}(0) = \mathfrak{f}\_{\lambda}^{(q) \prime}(0) = q\_{\prime} \qquad \overline{\mathcal{Q}}\_{\lambda}^{(q) \prime \prime}(\mathbf{x}) = \mathfrak{f}\_{\lambda}^{(q) \prime \prime}(0) = q\_{\prime} \quad \mathbf{x} \in \mathbb{R};$$

• for max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1} (mostly but not only for *<sup>λ</sup>* <sup>∈</sup> <sup>R</sup>\[0, 1]) (lower bound):

$$\underline{Q}\_{\lambda}^{(q)}(0) = \mathfrak{f}\_{\lambda}^{(q)}(0) = q - \beta\_{\lambda \prime} \qquad \underline{Q}\_{\lambda}^{(q) \prime}(0) = \mathfrak{f}\_{\lambda}^{(q) \prime}(0) = q\_{\prime} \qquad \underline{Q}\_{\lambda}^{(q) \prime \prime}(\ge) = \mathfrak{f}\_{\lambda}^{(q) \prime \prime}(0) = q\_{\prime} \quad \ge \in \mathbb{R};$$

• for max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1} in combination with (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*) <sup>2</sup> <sup>−</sup> <sup>2</sup> · *<sup>q</sup>* · (*<sup>q</sup>* <sup>−</sup> *<sup>β</sup>λ*) <sup>≥</sup> <sup>0</sup> (mostly but not only for *λ* ∈ R\[0, 1]) (upper bound):

$$\overline{\mathcal{Q}}\_{\lambda}^{(q)}(0) = \mathfrak{f}\_{\lambda}^{(q)}(0) = q - \beta\_{\lambda'} \quad \overline{\mathcal{Q}}\_{\lambda}^{(q) \prime}(0) = \mathfrak{f}\_{\lambda}^{(q) \prime}(0) = q \quad \overline{\mathcal{Q}}\_{\lambda}^{(q) \prime \prime}(\mathfrak{x}) = \mathfrak{f}\_{\lambda}^{(q) \prime \prime}(-\log(q)) = 1, \quad \mathfrak{x} \in \mathbb{R}.$$

If max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1} and (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*) <sup>2</sup> <sup>−</sup> <sup>2</sup> · *<sup>q</sup>* · (*<sup>q</sup>* <sup>−</sup> *<sup>β</sup>λ*) <sup>&</sup>lt; 0, then a real-valued solution *Q* (*q*) *λ* (*x*) = *x* does not exist and we set *x* (*q*) 0 := *x* (*q*) 0 := − log(*q*), with *ξ* (*q*)0 *λ x* (*q*) 0 = 1. The above considerations lead to corresponding unique choices of constants *a* (*q*) *λ* , *b* (*q*) *λ* , *c* (*q*) *λ* , *a* (*q*) *λ* , *b* (*q*) *λ* , *c* (*q*) *λ* culminating in

$$\underline{Q}\_{\lambda}^{(q)}(\mathbf{x}) \ := \begin{cases} \frac{q}{2} \cdot \mathbf{c}^{\underline{\mathbf{f}}^{(q)}} \cdot \mathbf{x}^2 + q \cdot \mathbf{x} + q - \beta\_{\lambda}, & \text{if } \ 0 < q < \beta\_{\lambda} \ \text{(127)} \\\\ \frac{q}{2} \cdot \mathbf{x}^2 + q \cdot \mathbf{x} + q - \beta\_{\lambda}, & \text{if } \ \max\{0, \beta\_{\lambda}\} < q < \min\{1, e^{\beta\_{\lambda} - 1}\} \ \text{(128)} \\\\ \frac{q}{2} \cdot \mathbf{x}^2 + q \cdot \mathbf{x} + q - \beta\_{\lambda}, & \text{if } \ 0 < q < \beta\_{\lambda} \ \text{(128)} \\\\ \frac{1}{2} \cdot \mathbf{x}^2 + q \cdot \mathbf{x} + q - \beta\_{\lambda}, & \text{if } \ \max\{0, \beta\_{\lambda}\} < q < \min\{1, e^{\beta\_{\lambda} - 1}\} \ \text{.} \end{cases} \tag{128}$$

#### *6.6. Closed-Form Bounds for Power Divergences of Non-Kullback-Leibler-Information-Divergence Type*

Analogously to Section 4 (see especially Section 4.1), for orders *λ* ∈ R\{0, 1} all the results of the previous Sections 6.1–6.5 carry correspondingly over from closed-form bounds of the Hellinger integrals *Hλ*(·||·) to closed-form bounds of the total variation distance *V*(·||·), by virtue of the relation (cf. (12))

$$2\left(1 - H\_{\frac{1}{2}}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n})\right) \le V(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) \le 2\sqrt{1 - \left(H\_{\frac{1}{2}}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n})\right)^2} \le 2$$

to closed-form bounds of the Renyi divergences *Rλ*(·||·), by virtue of the relation (cf. (7))

$$0 \le \mathcal{R}\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) \\
= \frac{1}{\lambda(\lambda - 1)} \log H\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) \quad \text{with } \log 0 := -\infty$$

as well as to closed-form bounds of the power divergences *I<sup>λ</sup>* (·||·), by virtue of the relation (cf. (2))

$$I\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) = \frac{1 - H\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n})}{\lambda \cdot (1 - \lambda)}, \qquad n \in \mathbb{N}.$$

For the sake of brevity, the–merely repetitive–exact details are omitted.

#### *6.7. Applications to Decision Making*

The above-mentioned investigations of the Sections 6.1 to 6.6 can be applied to the context of Section 2.5 on *dichotomous* decision making on the space of all possible path scenarios (path space) of Poissonian Galton-Watson processes without (with) immigration GW(I) (e.g., in combination with our running-example epidemiological context of Section 2.3). More detailed, for the minimal mean decision loss (Bayes risk) R*<sup>n</sup>* defined by (18) we can derive explicit closed-form upper (respectively lower) bounds by using (19) respectively (20) together with the results of the Sections 6.1–6.5 concerning Hellinger integrals of order *λ* ∈ ]0, 1[; we can proceed analogously in the Neyman-Pearson context in order to deduce closed-form bounds of type II error probabilities, by means of (23) and (24). Moreover, in an analogous way we can employ the investigations of Section 6.6 on power divergences in order to obtain closed-form bounds of (i) the corresponding (cf. (21)) *weighted-average* decision risk reduction (weighted-average statistical information measure) about the degree of evidence deg concerning the parameter *θ* that can be attained by observing the GW(I)-path X*<sup>n</sup>* until stage *n*, as well as (ii) the corresponding (cf. (22)) *limit* decision risk reduction (limit statistical information measure). For the sake of brevity, the–merely repetitive–exact details are omitted.

#### **7. Hellinger Integrals and Power Divergences of Galton-Watson Type Diffusion Approximations**

#### *7.1. Branching-Type Diffusion Approximations*

One can show that a properly rescaled Galton-Watson process without (respectively with) immigration GW(I) converges weakly to a diffusion process *X*e := n *X*e*s* ,*s* ∈ [0, ∞[ o which is the unique, strong, nonnegative – and in case of *<sup>η</sup> σ* <sup>2</sup> <sup>≥</sup> <sup>1</sup> 2 strictly positive– solution of the stochastic differential equation (SDE) of the form

$$d\tilde{X}\_s = \left(\eta - \kappa \tilde{X}\_s\right)ds + \sigma \sqrt{\tilde{X}\_s} \, dW\_s \quad s \in [0, \infty[ , \qquad \tilde{X}\_0 \in ]0, \infty[ \text{ given} \,\text{s} \tag{129}$$

where *<sup>η</sup>* <sup>∈</sup> [0, <sup>∞</sup>[, *<sup>κ</sup>* <sup>∈</sup> [0, <sup>∞</sup>[, *<sup>σ</sup>* <sup>∈</sup>]0, <sup>∞</sup>[ are constants and *W<sup>s</sup>* , *s* ∈ [0, ∞[ denotes a standard Brownian motion with respect to the underlying probability measure *P*; see e.g., Feller [130], Jirina [131], Lamperti [132,133], Lindvall [134,135], Grimvall [136], Jagers [56], Borovkov [137], Ethier & Kurtz [138], Durrett [139] for the non-immigration case corresponding to *η* = 0, *κ* ≥ 0, Kawazu & Watanabe [140], Wei & Winnicki [141], Winnicki [64] for the immigration case corresponding to *η* 6= 0, *κ* = 0, as well as Sriram [142] for the general case *η* ∈ [0, ∞[, *κ* ∈ R. Feller-type branching processes of the form (129), which are special cases of continuous state branching processes with immigration (see e.g., Kawazu & Watanabe [140], Li [143], as well as Dawson & Li [144] for imbeddings to affine processes) play

for instance an important role in the modelling of the term structure of interest rates, cf. the seminal Cox-Ingersoll-Ross CIR model [145] and the vast follow-up literature thereof. Furthermore, (129) is also prominently used as (a special case of) Cox & Ross's [146] constant elasticity of variance CEV asset price process, as (part of) Heston's [147] stochastic asset-volatility framework, as a model of neuron activity (see e.g., Lansky & Lanska [148], Giorno et al. [149], Lanska et al. [150], Lansky et al [151], Ditlevsen & Lansky [152], Höpfner [153], Lansky & Ditlevsen [154]), as a time-dynamic description of the nitrous oxide emission rate from the soil surface (see e.g., Pedersen [155]), as well as a model for the individual hazard rate in a survival analysis context (see e.g., Aalen & Gjessing [156]).

Along these lines of branching-type diffusion limits, it makes sense to consider the solutions of two SDEs (129) with different fixed parameter sets (*η*, *κ*A, *σ*) and (*η*, *κ*H, *σ*), determine for each of them a corresponding approximating GW(I), investigate the Hellinger integral between the laws of these two GW(I), and finally calculate the limit of the Hellinger integral (bounds) as the GW(I) approach their SDE solutions. Notice that for technicality reasons (which will be explained below), the constants *η* and *σ* ought to be independent of A, H in our current context.

In order to make the above-mentioned limit procedure rigorous, it is reasonable to work with appropriate approximations such that in each convergence step *m* one faces the setup PNI ∪ PSP,1 (i.e., the non-immigration or the equal-fraction case), where the corresponding Hellinger integral can be calculated exactly in a recursive way, as stated in Theorem 1. Let us explain the details in the following.

Consider a sequence of GW(I) *X* (*m*) *m*∈N with probability laws *P* (*m*) • on a measurable space (Ω, F), where as above the subscript • stands for either the hypothesis H or the alternative A. Analogously to (1), we use for each fixed step *m* ∈ N the representation *X* (*m*) := n *X* (*m*) ` , ` ∈ N o with

$$X\_{\ell}^{(m)} := \sum\_{j=1}^{X\_{\ell-1}^{(m)}} Y\_{\ell-1,j}^{(m)} + \widetilde{Y}\_{\ell}^{(m)}, \qquad \ell \in \mathbb{N}, \qquad X\_0^{(m)} \in \mathbb{N} \text{ given}, \tag{130}$$

where under the law *P* (*m*) •


From arbitrary drift-parameters *η* ∈ [0, ∞[, *κ*• ∈ [0, ∞[, and diffusion-term-parameter *σ* > 0, we construct the offspring-distribution-parameter and the immigration-distribution parameter of the sequence *X* (*m*) ` `∈N by

$$\beta^{(m)}\_{\bullet} := 1 - \frac{\kappa\_{\bullet}}{\sigma^2 m} \qquad \text{and} \qquad \alpha^{(m)}\_{\bullet} := \beta^{(m)}\_{\bullet} \cdot \frac{\eta}{\sigma^2}. \tag{131}$$

Here and henceforth, we always assume that the approximation step *m* is large enough to ensure that *β* (*m*) • ∈]0, 1] and at least one of *β* (*m*) A , *β* (*m*) H is strictly less than 1; this will be abbreviated by *m* ∈ N. Let us point out that – as mentioned above–our choice entails the best-to-handle setup PNI ∪ PSP,1 (which does not happen if instead of *η* one uses *η*• with *η*<sup>A</sup> 6= *η*H). Based on the GW(I) *X* (*m*) , let us construct the *continuous-time* branching process *<sup>X</sup>*e(*m*) := n *X*e (*m*) *s* , *s* ∈ [0, ∞[ o by

$$
\widetilde{X}\_s^{(m)} := \frac{1}{m} X\_{\left\lfloor \sigma^2 m s \right\rfloor}^{(m)} \,'\,\tag{132}
$$

living on the state space *E* (*m*) := <sup>1</sup> *<sup>m</sup>* <sup>N</sup>0. Notice that *<sup>X</sup>*e(*m*) is constant on each time-interval <sup>h</sup> *k σ* <sup>2</sup>*m* , *k*+1 *σ* <sup>2</sup>*m* h and takes at *s* = *<sup>k</sup> σ* <sup>2</sup>*m* the value <sup>1</sup> *<sup>m</sup> X* (*m*) *k* of the *k*-th GW(I) generation size, divided by *m*, i.e., it "jumps" with the jump-size <sup>1</sup> *m X* (*m*) *<sup>k</sup>* − *X* (*m*) *k*−1 which is equal to the <sup>1</sup> *m* -fold difference to the previous generation size. From (132) one can immediately see the necessity of having *σ* to be independent of A, H because for the required law-equivalence in (the corresponding version of) (13) both models at stake have to "live" on the same time-scale *τ* (*m*) *s* := *σ* <sup>2</sup>*ms* . For this setup, one obtains the following convergenc result:

**Theorem 10.** *Let <sup>η</sup>* <sup>∈</sup> [0, <sup>∞</sup>[*, <sup>κ</sup>*• <sup>∈</sup> [0, <sup>∞</sup>[*, <sup>σ</sup>* <sup>∈</sup>]0, <sup>∞</sup>[ *and <sup>X</sup>*e(*m*) *be as defined in (130) to (132). Furthermore, let us suppose that* lim*m*→<sup>∞</sup> 1 *<sup>m</sup> X* (*m*) <sup>0</sup> = *X*e<sup>0</sup> > 0 *and denote by D*([0, ∞[, [0, ∞[) *the space of right-continuous functions <sup>f</sup>* : [0, <sup>∞</sup>[7→ [0, <sup>∞</sup>[ *with left limits. Then the sequence of processes <sup>X</sup>*e(*m*) *m*∈N *convergences in distribution in D*([0, ∞[, [0, ∞[) *to a diffusion process X*e *which is the unique strong, nonnegative–and in case of η σ* <sup>2</sup> <sup>≥</sup> <sup>1</sup> 2 *strictly positive–solution of the SDE*

$$d\tilde{X}\_s = \begin{pmatrix} \eta - \kappa\_\bullet \tilde{X}\_s \end{pmatrix} ds + \sigma \sqrt{\tilde{X}\_s} \, d\mathcal{W}\_s^\bullet \quad s \in [0, \infty], \qquad \tilde{X}\_0 \in ]0, \infty[ \, given \, \iota \tag{133}$$

*where W*• *s* , *s* ∈ [0, ∞[ *denotes a standard Brownian motion with respect to the limit probability measure P*e•*.*

**Remark 8.** *Notice that the condition <sup>η</sup> σ* <sup>2</sup> <sup>≥</sup> <sup>1</sup> 2 *can be interpreted in our approximation setup* (131) *as α* (*m*) • ≥ *β* (*m*) • /2*, which quantifies the intuitively reasonable indication that if the probability P*•[*Y*e (*m*) ` = 0] = *e* −*α* (*m*) • *of having no immigration is small enough relative to the probability P*•[*Y* (*m*) `−1,*<sup>k</sup>* <sup>=</sup> <sup>0</sup>] = *<sup>e</sup>* −*β* (*m*) • *of having no offspring (m* ∈ N*), then the limiting diffusion X never hits zero almost surely.* e

The corresponding proof of Theorem 10–which is outlined in Appendix A.4–is an adaption of the proof of Theorem 9.1.3 in Ethier & Kurtz [138] which deals with drift-parameters *η* = 0, *κ*• = 0 in the SDE (133) whose solution is approached on a *σ*−independent time scale by a sequence of (critical) Galton-Watson processes without immigration but with general offspring distribution with mean 1 and variance *σ*. Notice that due to (131) the latter is inconsistent with our Poissonian setup, but this is compensated by our chosen *σ*−dependent time scale. Other limit investigations for (133) involving offspring/immigration distributions and parametrizations which are also incompatible to ours, are e.g., treated in Sriram [142].

As illustration of our proposed approach, let us give the following

**Example 3.** *Consider the parameter setup* (*η*, *κ*•, *σ*) = (5, 2, 0.4) *and initial generation size X*e<sup>0</sup> = 3*. Figure 4 shows the diffusion-approximation X*e (*m*) *s (blue) of the corresponding solution X*e*<sup>s</sup> of the SDE* (133) *up to the time horizon T* = 10*, for the approximation steps m* ∈ {13, 50, 200, 1000}*. Notice that in this setup there holds* N = {*k* ∈ N : *k* ≥ 13} *(recall that* N *is the subset of the positive integers such that β* (*m*) • = 1 − *κ*• *σ* <sup>2</sup>·*<sup>m</sup>* > 0*). The "long-term mean" of the limit process X*e*<sup>s</sup> is <sup>η</sup> κ*• = 2.5 *and is indicated as red line. The "long-term mean" of the approximations X*e (*m*) *s is equal to <sup>α</sup>* (*m*) • 1−*β* (*m*) • = *η κ*• − *η σ* <sup>2</sup>·*<sup>m</sup>* = 2.5 − 31.25/*m and is displayed as green line.*

**Figure 4.** Simulation of the process *X*e (*m*) *s* for the approximation steps *m* ∈ {13, 50, 200, 1000} in the parameter setup (*η*, *κ*•, *σ*) = (5, 2, 0.4) and with initial starting value *X*e<sup>0</sup> = 3.

#### *7.2. Bounds of Hellinger Integrals for Diffusion Approximations*

For each approximation step *m* and each observation horizon *t* ∈ [0, ∞[, let us now investigate the behaviour of the Hellinger integrals *H<sup>λ</sup> P* (*m*),*CDA* A,*t P* (*m*),*CDA* H,*t* , where *P* (*m*),*CDA* •,*t* denotes the canonical law (under <sup>H</sup> resp. <sup>A</sup>) of the *continuous-time diffusion approximation <sup>X</sup>*e(*m*) (cf. (132)), restricted to [0, *t*]. It is easy to see that *H<sup>λ</sup> P* (*m*),*CDA* A,*t P* (*m*),*CDA* H,*t* coincides with *H<sup>λ</sup> P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c of the law restrictions of the GW(I) generations sizes *X* (*m*) ` `∈{0,...,b*<sup>σ</sup>* <sup>2</sup>*mt*c} , where b*<sup>σ</sup>* <sup>2</sup>*mt*c *σ* <sup>2</sup>*m* can be interpreted as the last "jump-time" of *<sup>X</sup>*e(*m*) before *<sup>t</sup>*. These Hellinger integrals obey the results of


In order to obtain the desired Hellinger integral limits lim*m*→<sup>∞</sup> *H<sup>λ</sup> P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c , one faces several technical problems which will be described in the following. To begin with, for fixed *m* ∈ N we apply the Propositions 2(b), 3(b), 4(b), 5(b) to the current setup (*β* (*m*) A , *β* (*m*) H , *α* (*m*) A , *α* (*m*) H ) ∈ PNI ∪ PSP,1 with

$$\beta^{(m)}\_{\bullet} := \beta\_{\bullet}(m, \kappa\_{\bullet}, \sigma^{2}) \; := 1 - \frac{\kappa\_{\bullet}}{\sigma^{2}m} \qquad \text{and} \quad a^{(m)}\_{\bullet} := a\_{\bullet}(m, \kappa\_{\bullet}, \sigma^{2}, \eta) \; := \beta^{(m)}\_{\bullet} \cdot \frac{\eta}{\sigma^{2}} \quad \text{(cf. (131))}.$$

Notice that *η* = 0 corresponds to the no-immigration (NI) case and that *<sup>α</sup>* (*m*) • *β* (*m*) • = *η σ* 2 . Accordingly, we set *α* (*m*) *λ* := *λ* · *α* (*m*) <sup>A</sup> + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*) · *<sup>α</sup>* (*m*) H , *β* (*m*) *λ* := *λ* · *β* (*m*) <sup>A</sup> + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*) · *<sup>β</sup>* (*m*) H . By using

$$q\_{\lambda}^{(m)} := \left. q(m, \kappa\_{\bullet}, \sigma^2, \lambda) \right| := \left. \left( \mathcal{J}\_{\mathcal{A}}^{(m)} \right)^{\lambda} \left( \mathcal{J}\_{\mathcal{H}}^{(m)} \right)^{1-\lambda} \right|, \qquad \lambda \in \mathbb{R} \backslash \{0, 1\}, \tag{134}$$

as well as the connected sequence *a* (*m*) *n n*∈N := *a* (*q* (*m*) *λ* ) *n n*∈N we arrive at the

**Corollary 13.** *For all β* (*m*) A , *β* (*m*) H , *α* (*m*) A , *α* (*m*) H , *λ* ∈ (P*NI* ∪ P*SP,1*) × (R\{0, 1}) *and all population sizes X* (*m*) <sup>0</sup> ∈ N *there holds*

$$h\_{\lambda}^{(m)} := H\_{\lambda} \left( P\_{\mathcal{A}\_{\left\lfloor \sigma^{2}mt \right\rfloor}}^{(m)} \left| \left| P\_{\mathcal{H}\_{\left\lfloor \sigma^{2}mt \right\rfloor}}^{(m)} \right. \right. \right) \\ = \exp \left\{ a\_{\left\lfloor \sigma^{2}mt \right\rfloor}^{(q\_{\lambda}^{(m)})} \cdot X\_{0}^{(m)} + \frac{\eta}{\sigma^{2}} \sum\_{k=1}^{\left\lfloor \sigma^{2}mt \right\rfloor} a\_{k}^{(q\_{\lambda}^{(m)})} \right\} \tag{135}$$

*with η* = 0 *in the NI case.*

In the following, we employ the SDE-parameter constellations (which are consistent with (131) in combination with our requirement to work here only on (PNI ∪ PSP,1))

$$\tilde{\mathcal{P}}\_{\mathrm{NI}} \quad := \left\{ (\mathbb{x}\_{\mathcal{A}}, \mathbb{x}\_{\mathcal{H}}, \eta)\_{\prime}, \eta = 0, \,\,\mathbb{x}\_{\mathcal{A}} \in [0, \infty[ \, , \,\mathbb{x}\_{\mathcal{H}} \in [0, \infty[ \, , \,\mathbb{x}\_{\mathcal{A}} \neq \kappa\_{\mathcal{H}} \,] \, ]\, ]\, ]\, \right. \tag{136}$$

$$\tilde{\mathcal{P}}\_{\text{SP},1} \quad := \left\{ (\mathbb{x}\_{\mathcal{A}}, \mathbb{x}\_{\mathcal{H}}, \eta), \,\eta > 0, \,\,\mathbb{x}\_{\mathcal{A}} \in [0, \infty[ \, , \,\mathbb{x}\_{\mathcal{H}} \in [0, \infty[ \, , \,\mathbb{x}\_{\mathcal{A}} \neq \mathsf{x}\_{\mathcal{H}} \right] \,. \tag{137}$$

Due to the–not in closed-form representable–recursive nature of the sequences *a* (*q*) *n n*∈N defined by (36), the calculation of lim*m*→<sup>∞</sup> *h* (*m*) *λ* in (135) seems to be not (straightforwardly) tractable; after all, one "has to move along" a *sequence* of recursions (roughly speaking) since *σ* <sup>2</sup>*mt* → ∞ as *m* tends to infinity. One way to "circumvent" such technical problems is to compute instead of the limit lim*m*→<sup>∞</sup> *h* (*m*) *λ* of the (exact values of the) Hellinger integrals *h* (*m*) *λ* , the limits of the corresponding (explicit) closed-form lower resp. upper bounds adapted from Theorem 5. In order to achieve this, one first needs a preparatory step, due to the fact that the sequence *a* (*q* (*m*) *λ* ) b*σ* <sup>2</sup>*mt*c *m*∈N (and hence its bounds leading to closed-form expressions) does not necessarily converge for all *λ* ∈ R\[0, 1]; roughly, this can be conjectured from the Propositions 3(c) and 5(c) in combination with *σ* <sup>2</sup>*mt* → ∞. Correspondingly, for our "sequence-of-recursions" context equipped with the diffusion-limit's drift-parameter constellations (*κ*A, *κ*H, *η*) we have to derive a "convergence interval" [e*λ*−, e*λ*+]\[0, 1] which replaces the single-recursion-concerning [*λ*−, *λ*+]\[0, 1] (cf. Lemma 1). This amounts to

**Proposition 15.** *For all* (*κ*A, *κ*H, *η*) ∈ Pe*NI* ∪ Pe *SP,1 define*

$$\begin{array}{rclclclclcl} 0 & \sim \widetilde{\lambda}\_{-} & := & \begin{cases} -\infty, & \text{if } & \text{\bf x}\_{\mathcal{A}} < \kappa\_{\mathcal{H}}, \\ -\frac{\mathbf{x}\_{\mathcal{A}}^{2}}{\mathbf{x}\_{\mathcal{A}}^{2} - \mathbf{x}\_{\mathcal{A}}^{2}}, & \text{if } & \text{\bf x}\_{\mathcal{A}} > \kappa\_{\mathcal{H}}, \end{cases} & \text{and} & 1 < \widetilde{\lambda}\_{+} := & \begin{cases} \frac{\mathbf{x}\_{\mathcal{A}}^{2}}{\mathbf{x}\_{\mathcal{A}}^{2} - \mathbf{x}\_{\mathcal{A}}^{2}}, & \text{if } & \text{\bf x}\_{\mathcal{A}} < \kappa\_{\mathcal{H}}, \\ \infty, & \text{if } & \text{\bf x}\_{\mathcal{A}} > \kappa\_{\mathcal{H}}. \end{cases} \end{array} \tag{138}$$

*Then, for all* (*κ*A, *κ*H, *η*, *λ*) ∈ (Pe*NI* ∪ Pe *SP,1*) × ]e*λ*−, e*λ*+[ \ [0, 1] *there holds for all sufficiently large m* ∈ N

$$q\_{\lambda}^{(m)} := \left(1 - \frac{\kappa\_{\mathcal{A}}}{\sigma^2 m}\right)^{\lambda} \left(1 - \frac{\kappa\_{\mathcal{H}}}{\sigma^2 m}\right)^{1-\lambda} < \min\left\{1, \, e^{\oint\_{\lambda}^{(m)} -1}\right\},\tag{139}$$

*and thus the sequence a* (*q* (*m*) *λ* ) *n n*∈N *converges to the fixed point x*(*m*) 0 ∈ i 0, <sup>−</sup> log *q* (*m*) *λ* h*.*

This will be proved in Appendix A.4.

We are now in the position to determine bounds of the Hellinger integral limits lim*m*→<sup>∞</sup> *H<sup>λ</sup> P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c in form of *m*-limits of appropriate versions of closed-form bounds from Section 6. For the sake of brevity, let us henceforth use the abbreviations *x* (*m*) 0 := *x* (*q* (*m*) *λ* ) 0 , Γ (*m*) <sup>&</sup>lt; := Γ (*q* (*m*) *λ* ) <sup>&</sup>lt; = *q* (*m*) *λ* 2 · *e x* (*m*) <sup>0</sup> · *x* (*m*) 0 2 , Γ (*m*) <sup>&</sup>gt; := Γ (*q* (*m*) *λ* ) <sup>&</sup>gt; = *q* (*m*) *λ* 2 · *x* (*m*) 0 2 , *d* (*m*),*S* := *d* (*q* (*m*) *λ* ),*<sup>S</sup>* = *x* (*m*) <sup>0</sup> −(*q* (*m*) *<sup>λ</sup>* −*β* (*m*) *λ* ) *x* (*m*) 0 and *d* (*m*),*T* := *d* (*q* (*m*) *λ* ),*<sup>T</sup>* = *q* (*m*) *λ* · *e x* (*m*) <sup>0</sup> . By the above considerations, the Theorem 5 (together with Remark 7(a)) adapts to the current setup as follows:

**Corollary 14.** *(a) For all* (*κ*A, *κ*H, *η*, *λ*) ∈ (Pe*NI* ∪ Pe *SP*,1)×]0, 1[*, all t* ∈ [0, ∞[*, all approximation steps <sup>m</sup>* <sup>∈</sup> <sup>N</sup> *and all initial population sizes X*(*m*) <sup>0</sup> ∈ N *the Hellinger integral can be bounded by*

$$\begin{split} \mathbf{C}\_{\boldsymbol{\lambda}, \mathbf{X}\_{0}^{(m)}, t}^{(m), L} &:= \left\{ \mathbf{x}\_{0}^{(m)} \cdot \left[ \mathbf{X}\_{0}^{(m)} - \frac{\eta}{\sigma^{2}} \frac{d^{(m), T}}{1 - d^{(m), T}} \right] \left( 1 - \left( d^{(m), T} \right)^{\lfloor \sigma^{2} m t \rfloor} \right) \right. \\ &\left. + \underline{\zeta}\_{\lfloor \sigma^{2} m t \rfloor}^{(m)} \cdot \mathbf{X}\_{0}^{(m)} + \frac{\eta}{\sigma^{2}} \cdot \underline{\mathfrak{d}}\_{\lfloor \sigma^{2} m t \rfloor}^{(m)} \right\} \\ &\cdots \end{split} \tag{140}$$

$$\begin{split} \leq & \quad H\_{\lambda} \left( \left. P\_{\mathcal{A}, \left[ \sigma^{2} mt \right]}^{(m)} \right| \left| \left. P\_{\mathcal{H}, \left[ \sigma^{2} mt \right]}^{(m)} \right| \right) \\ \leq & \quad \exp \left\{ \left. \mathbf{x}\_{0}^{(m)} \cdot \left[ \mathbf{X}\_{0}^{(m)} - \frac{\eta}{\sigma^{2}} \frac{d^{(m),S}}{1 - d^{(m),S}} \right] \right| \left( 1 - \left( d^{(m),S} \right)^{\left[ \sigma^{2} mt \right]} \right) + \left. \mathbf{x}\_{0}^{(m)} \frac{\eta}{\sigma^{2}} \cdot \left| \left. \sigma^{2} mt \right| \right| \right. \\ \left. \left. \left. \mathbf{-}\_{\left[ \left. \sigma^{2} mt \right]}^{\left( \left. \mathbf{x}\_{0}^{(m)} \right| \right)} - \frac{\eta}{\sigma^{2}} \cdot \overline{\theta}\_{\left[ \left. \sigma^{2} mt \right|}^{\left( \left. \mathbf{x}\_{0}^{(m)} \right| \right)} \right. \right. \right. \right. \\ \left. \left. \left. \mathbf{-}\_{\left( \left. \mathbf{x}\_{0}^{(m)} \right| \right)}^{\left( \left. \mathbf{x}\_{0}^{(m)} \right| \right)} \right. \right. \right. \right. \right. \end{split} \tag{141}$$

*where we define analogously to* (98) *to* (101)

$$\underline{\zeta}\_{\rm m}^{(m)} := -\Gamma\_{<}^{(m)} \cdot \frac{\left(d^{(m),T}\right)^{n-1}}{1 - d^{(m),T}} \cdot \left(1 - \left(d^{(m),T}\right)^{n}\right) > 0 \,, \tag{142}$$

$$\underline{\underline{\mathcal{G}}}\_{n}^{(m)} := -\Gamma\_{<}^{(m)} \cdot \frac{1 - \left(d^{(m),T}\right)^{n}}{\left(1 - d^{(m),T}\right)^{2}} \cdot \left[1 - \frac{d^{(m),T}\left(1 + \left(d^{(m),T}\right)^{n}\right)}{1 + d^{(m),T}}\right] > 0\,,\tag{143}$$

$$\mathbb{E}\_{\mathbf{Z}}^{(m)} \quad := \quad \Gamma\_{<}^{(m)} \cdot \left[ \frac{\left(d^{(m),S}\right)^{n} - \left(d^{(m),T}\right)^{n}}{d^{(m),S} - d^{(m),T}} - \left(d^{(m),S}\right)^{n-1} \cdot \frac{1 - \left(d^{(m),T}\right)^{n}}{1 - d^{(m),T}} \right] > 0 \,, \tag{144}$$

$$\overline{\sigma}\_{n}^{(m)} \quad := \quad \Gamma\_{<}^{(m)} \cdot \frac{d^{(m),T}}{1 - d^{(m),T}} \cdot \left[ \frac{1 - \left(d^{(m),S}d^{(m),T}\right)^{n}}{1 - d^{(m),S}d^{(m),T}} - \frac{\left(d^{(m),S}\right)^{n} - \left(d^{(m),T}\right)^{n}}{d^{(m),S} - d^{(m),T}} \right] \\ > 0 \quad . \tag{145}$$

*Notice that* (140) *and* (141) *simplify significantly for* (*κ*A, *κ*H, *η*, *λ*) ∈ Pe*NI*×]0, 1[ *for which η* = 0 *holds. (b) For all* (*κ*A, *κ*H, *η*, *λ*) ∈ (Pe*NI* ∪ Pe *SP,1*) × e*λ*−, e*λ*<sup>+</sup> - \ [0, 1] *and all initial population sizes X* (*m*) <sup>0</sup> ∈ N *the* *Hellinger integral bounds* (140) *and* (141) *are valid for all sufficiently large m* ∈ N*, where the expressions* (142) *to* (145) *have to be replaced by*

$$\underline{\zeta}\_n^{(m)} \quad := \quad \Gamma\_>^{(m)} \cdot \frac{\left(d^{(m),T}\right)^n - \left(d^{(m),S}\right)^{2n}}{d^{(m),T} - \left(d^{(m),S}\right)^2} > 0 \,, \tag{146}$$

$$\underline{\mathcal{G}}\_{n}^{(m)} := \frac{\Gamma\_{>}^{(m)}}{d^{(m),T} - \left(d^{(m),S}\right)^{2}} \cdot \left[\frac{d^{(m),T} \cdot \left(1 - \left(d^{(m),T}\right)^{n}\right)}{1 - d^{(m),T}} - \frac{\left(d^{(m),S}\right)^{2} \cdot \left(1 - \left(d^{(m),S}\right)^{2n}\right)}{1 - \left(d^{(m),S}\right)^{2}}\right] > 0,$$

$$\underline{\mathcal{G}}\_{n}^{(m)} := \quad \Gamma\_{>}^{(m)} \cdot \left(d^{(m),S}\right)^{n-1} \cdot \left[n - \frac{1 - \left(d^{(m),T}\right)^{n}}{1 - d^{(m),T}}\right] > 0,\tag{147}$$

$$\overline{\theta}\_n^{(m)} := \quad \Gamma\_{>}^{(m)} \cdot \left[ \frac{d^{(m),S} - d^{(m),T}}{\left(1 - d^{(m),S}\right)^2 \left(1 - d^{(m),T}\right)} \cdot \left(1 - \left(d^{(m),S}\right)^n\right)^n \right.\tag{148}$$

$$\left[1+\frac{d^{(m),T}\left(1-\left(d^{(m),S}d^{(m),T}\right)^{n}\right)}{\left(1-d^{(m),T}\right)\left(1-d^{(m),S}d^{(m),T}\right)}-\frac{\left(d^{(m),S}\right)^{n}}{1-d^{(m),S}}\cdot n\right].\tag{149}$$

Let us finally present the desired assertions on the limits of the bounds given in Corollary 14 as the approximation step *m* tends to infinity, by employing for *λ* ∈ e*λ*−, e*λ*<sup>+</sup> - ! [0, 1] the quantities

$$
\kappa\_{\lambda} := \lambda \kappa\_{\mathcal{A}} + (1 - \lambda) \kappa\_{\mathcal{H}} \qquad \text{as well as} \qquad \Lambda\_{\lambda} := \sqrt{\lambda \kappa\_{\mathcal{A}}^2 + (1 - \lambda) \kappa\_{\mathcal{H}}^2} \tag{150}
$$

for which the following relations hold:

$$
\Lambda\_{\lambda} > \kappa\_{\lambda} > 0,\qquad\qquad\text{for}\quad\lambda \in \left]0,1\right[\,,\tag{151}
$$

$$0 < \Lambda\_{\lambda} < \kappa\_{\lambda'} \qquad \qquad \text{for} \quad \lambda \in \left[\tilde{\lambda}\_{-}, \tilde{\lambda}\_{+}\right] \backslash \left[0, 1\right]. \tag{152}$$

**Theorem 11.** *Let the initial SDE-value X*e<sup>0</sup> ∈]0, ∞[ *be arbitrary but fixed, and suppose that* lim*m*→<sup>∞</sup> 1 *<sup>m</sup> X* (*m*) <sup>0</sup> = *X*e0*. Then, for all* (*κ*A, *κ*H, *η*, *λ*) ∈ (Pe*NI* ∪ Pe *SP*,1) × e*λ*<sup>−</sup> , e*λ*<sup>+</sup> - \ {0, 1} *and all t* ∈ [0, ∞[ *the Hellinger integral limit can be bounded by*

$$\begin{split} \mathcal{D}^{L}\_{\lambda,\widetilde{\chi}\_{0,t}} &:= \quad \exp\left\{-\frac{\Lambda\_{\lambda}-\kappa\_{\lambda}}{\sigma^{2}} \left[\widetilde{\chi}\_{0}-\frac{\eta}{\Lambda\_{\lambda}}\right] \left(1-e^{-\Lambda\_{\lambda}t}\right) - \frac{\eta}{\sigma^{2}} \left(\Lambda\_{\lambda}-\kappa\_{\lambda}\right) \cdot t \\ &\quad + L^{(1)}\_{\lambda}(t) \cdot \widetilde{\mathcal{X}}\_{0} + \frac{\eta}{\sigma^{2}} \cdot L^{(2)}\_{\lambda}(t) \right\} \\ &\leq \quad \lim\_{m \to \infty} H\_{\lambda} \left(P^{(m)}\_{\mathcal{A},\left[e^{2\pi m}\right]} \left|\left|\left|P^{(m)}\_{\mathcal{H},\left[e^{2\pi m}\right]}\right|\right.\right. \\ &\leq \quad \exp\left\{-\frac{\Lambda\_{\lambda}-\kappa\_{\lambda}}{\sigma^{2}} \left[\widetilde{\mathcal{X}}\_{0} - \frac{\eta}{\frac{1}{2}(\Lambda\_{\lambda}+\kappa\_{\lambda})}\right] \left(1-e^{-\frac{1}{2}(\Lambda\_{\lambda}+\kappa\_{\lambda})\cdot t}\right) - \frac{\eta}{\sigma^{2}} \left(\Lambda\_{\lambda}-\kappa\_{\lambda}\right) \cdot t \\ &\quad - L^{(1)}\_{\lambda}(t) \cdot \widetilde{\mathcal{X}}\_{0} - \frac{\eta}{\sigma^{2}} \cdot L^{(2)}\_{\lambda}(t) \right) \quad =: \quad D^{\mathcal{U}}\_{\lambda,\widetilde{\mathcal{X}}\_{0,t}}. \end{split} \tag{154}$$

*where for the (sub)case of all λ* ∈]0, 1[ *and all t* ≥ 0

$$L\_{\lambda}^{(1)}(t) \quad := \quad \frac{\left(\Lambda\_{\mathbb{N}} - \kappa\_{\lambda}\right)^{2}}{2\sigma^{2} \cdot \Lambda\_{\mathbb{N}}} \cdot e^{-\Lambda\_{\mathbb{N}} \cdot t} \cdot \left(1 - e^{-\Lambda\_{\mathbb{N}} \cdot t}\right) \,, \tag{155}$$

$$L\_{\lambda}^{(2)}(t) \quad := \frac{1}{4} \cdot \left(\frac{\Lambda\_{\mathbb{k}} - \kappa\_{\lambda}}{\Lambda\_{\mathbb{k}}}\right)^2 \cdot \left(1 - e^{-\Lambda\_{\mathbb{k}} \cdot t}\right)^2,\tag{156}$$

$$\mathcal{U}^{(1)}\_{\lambda}(t) \quad := \quad \frac{(\Lambda\_{\mathbb{K}} - \kappa\_{\lambda})^2}{\sigma^2} \cdot \left[ \frac{e^{-\frac{1}{2}(\Lambda\_{\mathbb{K}} + \kappa\_{\lambda}) \cdot t} - e^{-\Lambda\_{\mathbb{K}} \cdot t}}{\Lambda\_{\mathbb{K}} - \kappa\_{\lambda}} - \frac{e^{-\frac{1}{2}(\Lambda\_{\mathbb{K}} + \kappa\_{\lambda}) \cdot t} \left(1 - e^{-\Lambda\_{\mathbb{K}} \cdot t}\right)}{2 \cdot \Lambda\_{\mathbb{K}}} \right],\tag{157}$$

$$\mathcal{U}\_{\lambda}^{(2)}(t) \quad := \quad \frac{(\Lambda\_{\mathrm{h}} - \kappa\_{\lambda})^{2}}{\Lambda\_{\mathrm{h}}} \cdot \left[ \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\mathrm{h}} + \kappa\_{\lambda}) \cdot t}}{3\Lambda\_{\mathrm{h}} + \kappa\_{\lambda}} + \frac{e^{-\Lambda\_{\mathrm{h}} \cdot t} - e^{-\frac{1}{2}(\Lambda\_{\mathrm{h}} + \kappa\_{\lambda}) \cdot t}}{\Lambda\_{\mathrm{h}} - \kappa\_{\lambda}} \right], \tag{158}$$

*and for the remaining (sub)case of all λ* ∈ e*λ*−, e*λ*<sup>+</sup> -[0, 1] *and all t* <sup>≥</sup> <sup>0</sup>

$$L\_{\lambda}^{(1)}(t) \quad := \quad \frac{\left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)^{2}}{2\sigma^{2} \cdot \kappa\_{\lambda}} \cdot e^{-\Lambda\_{\lambda} \cdot t} \cdot \left(1 - e^{-\kappa\_{\lambda} \cdot t}\right) \,, \tag{159}$$

$$L\_{\lambda}^{(2)}(t) \quad := \quad \frac{\left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)^{2}}{2 \cdot \kappa\_{\lambda}} \cdot \left[\frac{1 - e^{-\Lambda\_{\lambda} \cdot t}}{\Lambda\_{\lambda}} - \frac{1 - e^{-\left(\Lambda\_{\lambda} + \kappa\_{\lambda}\right) \cdot t}}{\Lambda\_{\lambda} + \kappa\_{\lambda}}\right] \,, \tag{160}$$

$$\mathcal{U}\_{\lambda}^{(1)}(t) \quad := \frac{\left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)^{2}}{2 \cdot \sigma^{2}} \cdot e^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t} \cdot \left[t - \frac{1 - e^{-\Lambda\_{\lambda} \cdot t}}{\Lambda\_{\lambda}}\right],\tag{161}$$

$$\mathcal{U}\_{\lambda}^{(2)}(t) \quad := \quad (\Lambda\_{\lambda} - \kappa\_{\lambda})^2 \cdot \left[ \frac{\left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)\left(1 - e^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}\right)}{\Lambda\_{\lambda} \cdot \left(\Lambda\_{\lambda} + \kappa\_{\lambda}\right)^2} + \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}}{\Lambda\_{\lambda} \cdot \left(3\Lambda\_{\lambda} + \kappa\_{\lambda}\right)} - \frac{e^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}}{\Lambda\_{\lambda} + \kappa\_{\lambda}} \cdot t \right]. \tag{162}$$

*Notice that the components L* (*i*) *λ* (*t*) *and U* (*i*) *λ* (*t*) *for i* = 1, 2 *and in both cases λ* ∈]0, 1[ *and λ* ∈ e*λ*−, e*λ*<sup>+</sup> - [0, 1] *are strictly positive for t* > 0 *and do not depend on the parameter η. Furthermore, the bounds D<sup>L</sup> λ*,*X*e 0,*t and D<sup>U</sup> λ*,*X*e 0,*t simplify significantly in the case* (*κ*A, *κ*H, *η*) ∈ Pe*NI, for which η* = 0 *holds.*

This will be proved in Appendix A.4. For the time-asymptotics, we obtain the

**Corollary 15.** *Let the initial SDE-value X*e<sup>0</sup> ∈]0, ∞[ *be arbitrary but fixed, and suppose that* lim*m*→<sup>∞</sup> 1 *<sup>m</sup> X* (*m*) <sup>0</sup> = *X*e0*. Then:*

*(a) For all* (*κ*A, *κ*H, *η*, *λ*) ∈ Pe*NI* × e*λ*−, e*λ*<sup>+</sup> - \{0, 1} *the Hellinger integral limit converges to*

$$\lim\_{t \to \infty} \lim\_{m \to \infty} \log \left( H\_{\lambda} \left( P\_{\mathcal{A}, \left[ \sigma^{2} m \mathbb{I} \right]}^{(m)} \middle| \begin{aligned} & \left| P\_{\mathcal{H}, \left[ \sigma^{2} m \mathbb{I} \right]}^{(m)} \right. \right) \right) \right) &= \ -\frac{\widetilde{\mathcal{X}}\_{0}}{\sigma^{2}} \cdot \left( \Lambda\_{\lambda} - \kappa\_{\lambda} \right) \ & \left\{ \begin{array}{c} < 0, \quad f \text{or} \quad \lambda \in ]0, 1[ \text{ }. \end{array} \right. \\\\ & > 0, \quad f \text{or} \quad \lambda \in ]\widetilde{\Lambda}\_{-} \widetilde{\Lambda}\_{+} \left[ \left\langle \begin{array}{c} 0, 1 \end{array} \right\rangle . \end{aligned} \right.$$

*(b) For all* (*κ*A, *κ*H, *η*, *λ*) ∈ Pe *SP*,1 × e*λ*−, e*λ*<sup>+</sup> - \{0, 1} *the Hellinger integral limit possesses the asymptotical behaviour*

$$\lim\_{t \to \infty} \frac{1}{t} \log \left( \lim\_{m \to \infty} H\_{\lambda} \left( \left. P\_{\mathcal{A}, \left[ \varepsilon^{2} m \right]}^{(m)} \right| \right| \left| P\_{\mathcal{H}, \left[ \varepsilon^{2} m \right]}^{(m)} \right) \right) \\ = -\frac{\eta}{\sigma^{2}} \cdot \left( \Lambda\_{\lambda} - \kappa\_{\lambda} \right) \quad \left\{ \begin{array}{c} < 0, \quad \text{for} \quad \lambda \in ]0, 1[\\\\> 0, \quad \text{for} \quad \lambda \in ]\widetilde{\Lambda}\_{-}, \widetilde{\Lambda}\_{+} \left[ \left. \left[ \left. \left[ 0, 1 \right] \right| \right| \right| \right] \end{array} \right)$$

The assertions of Corollary 15 follow immediately by inspecting the expressions in the exponential of (153) and (154) in combination with (155) to (162).

#### *7.3. Bounds of Power Divergences for Diffusion Approximations*

Analogously to Section 4 (see especially Section 4.1), for orders *λ* ∈ R\{0, 1} all the results of the previous Section 7.2 carry correspondingly over from (limits of) bounds of the Hellinger integrals *H<sup>λ</sup> P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c to (limits of) bounds of the total variation distance *V P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c (by virtue of (12)), to (limits of) bounds of the Renyi divergences *Rλ P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c (by virtue of (7)) as well as to (limits of) bounds of the power divergences *Iλ P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c (by virtue of (2)). For the sake of brevity, the–merely repetitive–exact details are omitted. Moreover, by combining the outcoming results on the above-mentioned power divergences with parts of the Bayesian-decision-making context of Section 2.5, we obtain corresponding assertions on (i) the (cf. (21)) *weighted-average* decision risk reduction (weighted-average statistical information measure) about the degree of evidence deg concerning the parameter *θ* that can be attained by observing the GWI-path X*<sup>n</sup>* until stage *n*, as well as (ii) the (cf. (22)) *limit* decision risk reduction (limit statistical information measure).

In the following, let us concentrate on the derivation of the Kullback-Leibler information divergence KL (relative entropy) within the current diffusion-limit framework. Notice that altogether we face two limit procedures simultaneously: by the first limit lim*λ*↑<sup>1</sup> *Iλ P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c ||*P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c we obtain the KL *I P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c ||*P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c for every fixed approximation step *m* ∈ N; on the other hand, for each fixed *λ* ∈]0, 1[, the second limit lim*m*→<sup>∞</sup> *I<sup>λ</sup> P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c ||*P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c describes the limit of the power divergence – as the sequence of rescaled and continuously interpolated GW(I)'s *X*e (*m*) *s s*∈[0,∞[ *m*∈N equipped with probability law *P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c resp. *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c up to time *σ* <sup>2</sup>*mt* converges weakly to the continuous-time CIR-type diffusion process *X*e*s s*∈[0,∞[ (with probability law *P*eA,*<sup>t</sup>* resp. *P*eH,*<sup>t</sup>* up to time *t*). In Appendix A.4 we shall prove that these two limits can be interchanged:

**Theorem 12.** *Let the initial SDE-value X*e<sup>0</sup> ∈]0, ∞[ *be arbitrary but fixed, and suppose that* lim*m*→<sup>∞</sup> 1 *<sup>m</sup> X* (*m*) <sup>0</sup> = *X*e0*. Then, for all* (*κ*A, *κ*H, *η*) ∈ Pe*NI* ∪ Pe *SP*,1 *and all t* ∈ [0, ∞[ *, one gets the Kullback-Leibler information divergence (relative entropy) convergences*

lim*m*→<sup>∞</sup> *I P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c <sup>=</sup> lim*m*→<sup>∞</sup> lim *λ*%1 *Iλ P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c = (*κ*A−*κ*H) 2 2*σ* <sup>2</sup>·*κ*<sup>A</sup> · h*X*e<sup>0</sup> <sup>−</sup> *η κ*A · 1 − *e* −*κ*A·*t* + *η* · *t* i , *if κ*<sup>A</sup> > 0, *κ* 2 H 2*σ* 2 · h *η* 2 · *t* <sup>2</sup> <sup>+</sup> *<sup>X</sup>*e<sup>0</sup> · *<sup>t</sup>* i , *if κ*<sup>A</sup> = 0, = lim *λ*%1 lim*m*→<sup>∞</sup> *Iλ P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c . (163)

This immediately leads to the following

**Corollary 16.** *Let the initial SDE-value X*e<sup>0</sup> ∈]0, ∞[ *be arbitrary but fixed, and suppose that* lim*m*→<sup>∞</sup> 1 *<sup>m</sup> X* (*m*) <sup>0</sup> = *X*e0*. Then, the KL limit* (163) *possesses the following time-asymptotical behaviour: (a) For all* (*κ*A, *κ*H, *η*) ∈ Pe*NI (i.e., η* = 0*) one gets*

$$\begin{array}{ccccc} \text{(i)} & \text{in the case } \kappa\_{\mathcal{A}} > 0 & \lim\_{t \to \infty} \lim\_{m \to \infty} I \left( P\_{\mathcal{A}, \left[ \sigma^{2} m \right]}^{(m)} \middle| \left| P\_{\mathcal{H}, \left[ \sigma^{2} m \right]}^{(m)} \right. \right) & = & \frac{\breve{X}\_{0} \cdot (\kappa\_{\mathcal{A}} - \kappa\_{\mathcal{H}})^{2}}{2\sigma^{2} \cdot \kappa\_{\mathcal{A}}},\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \Gamma\_{0} \cdot \sqrt{\kappa\_{\mathcal{A}}(m)} \end{array}$$

$$\text{I(ii)}\qquad\text{in the case }\kappa\_{\mathcal{A}}=0\qquad\lim\_{t\to\infty}\lim\_{m\to\infty}\frac{1}{t}\cdot I\left(P\_{\mathcal{A},\left[\sigma^{2}mt\right]}^{(m)}||P\_{\mathcal{H},\left[\sigma^{2}mt\right]}^{(m)}\right) = \frac{X\_{0}\cdot\kappa\_{\mathcal{H}}^{2}}{4\sigma^{2}}\dots$$

*(b) For all* (*κ*A, *κ*H, *η*) ∈ Pe *SP*,1 *(i.e., η* > 0*) one gets*

$$\begin{aligned} \text{(i)} \qquad &\text{in the case } \mathbb{K}\_{\mathcal{A}} > 0 \qquad \lim\_{t \to \infty} \lim\_{m \to \infty} \frac{1}{t} \cdot I\left(P\_{\mathcal{A}, \left[\sigma^{2}mt\right]}^{(m)} \, \middle|\, \left|P\_{\mathcal{H}, \left[\sigma^{2}mt\right]}^{(m)}\right|\right) = \frac{\eta \cdot (\mathbb{K}\_{\mathcal{A}} - \mathbb{K}\_{\mathcal{H}})^{2}}{2\sigma^{2} \cdot \mathbb{K}\_{\mathcal{A}}} \,\, \forall \,\, \forall m \in \mathbb{Z}\_{\omega} \end{aligned}$$

*(ii) in the case κ*<sup>A</sup> = 0 lim *t*→∞ lim*m*→<sup>∞</sup> 1 *t* 2 · *I P* (*m*) <sup>A</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c *P* (*m*) <sup>H</sup>,b*<sup>σ</sup>* <sup>2</sup>*mt*c = *η* · *κ* H 4*σ* 2 .

**Remark 9.** *In Appendix A.4 we shall see that the proof of the last (limit-interchange concerning) equality in* (163) *relies heavily on the use of the extra terms L* (1) *λ* (*t*), *L* (2) *λ* (*t*), *U* (1) *λ* (*t*), *U* (2) *λ* (*t*) *in* (153) *and* (154)*. Recall that these terms ultimately stem from (manipulations of) the corresponding parts of the "improved closed-form bounds" in Theorem 5, which were derived by using the linear inhomogeneous difference equations a* (*q*) *<sup>n</sup> resp. a* (*q*) *<sup>n</sup> (cf.* (92) *resp.* (94)*) instead of the linear homogeneous difference equations a* (*q*),*T <sup>n</sup> resp. a* (*q*),*S <sup>n</sup> (cf.* (78) *resp.* (79)*) as explicit approximates of the sequence a* (*q*) *<sup>n</sup> . Not only this fact shows the importance of this more tedious approach.*

Interesting comparisons of the above-mentioned results in Sections 7.2 and 7.3 with corresponding information measures of the solutions of the SDE (129) themselves (rather their branching approximations), can be found in Kammerer [157].

#### *7.4. Applications to Decision Making*

Analogously to Section 6.7, the above-mentioned investigations of the Sections 7.1–7.3 can be applied to the context of Section 2.5 on *dichotomous* decision making about GW(I)-type diffusion approximations of solutions of the stochastic differential Equation (129). For the sake of brevity, the–merely repetitive–exact details are omitted.

**Author Contributions:** Conceptualization, N.B.K. and W.S.; Formal analysis, N.B.K. and W.S.; Methodology, N.B.K. and W.S.; Visualization, N.B.K.; Writing, N.B.K. and W.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** Niels B. Kammerer received a scholarship of the "Studienstiftung des Deutschen Volkes" for his PhD Thesis.

**Acknowledgments:** We are very grateful to the referees for their patience to review this long manuscript, and for their helpful suggestions. Moreover, we would like to thank Andreas Greven for some useful remarks.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Proofs and Auxiliary Lemmas**

*Appendix A.1. Proofs and Auxiliary Lemmas for Section 3*

**Lemma A1.** *For all real numbers x*, *y*, *z* > 0 *and all λ* ∈ R *one has*

$$\left\{ \left. x^{\lambda} y^{1-\lambda} - \left( \lambda \, x z^{\lambda -1} + (1 - \lambda) \, y z^{\lambda} \right) \right\} \begin{cases} \leq 0, & \text{for } \lambda \in ]0, 1[ \\ = 0, & \text{for } \lambda \in \{0, 1\} \\ \geq 0, & \text{for } \lambda \in \mathbb{R} \backslash [0, 1] \end{cases} \right\}$$

*with equality in the cases <sup>λ</sup>* <sup>∈</sup> <sup>R</sup>\{0, 1} *iff <sup>x</sup> <sup>y</sup>* = *z.*

**Proof of Lemma A1.** For fixed *<sup>x</sup>*˜ :<sup>=</sup> *xzλ*−<sup>1</sup> <sup>&</sup>gt; 0, *<sup>y</sup>*˜ :<sup>=</sup> *yz<sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup> with *<sup>x</sup>*˜ <sup>6</sup><sup>=</sup> *<sup>y</sup>*˜ we inspect the function *g* on R defined by *g*(*λ*) := *x*˜ *λy*˜ <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> (*λx*˜ + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*)*y*˜) which satisfies *<sup>g</sup>*(0) = *<sup>g</sup>*(1) = 0, *g* 0 (0) = *y*˜ log(*x*˜/*y*˜) − (*x*˜ − *y*˜) < *y*˜((*x*˜/*y*˜) − 1) − (*x*˜ − *y*˜) = 0 and which is strictly convex. Thus, the assertion follows immediately by taking into account the obvious case *x*˜ = *y*˜.

**Proof of Properties 1.** Property (P9) is trivially valid. To show (P1) we assume 0 < *q* < *βλ*, which implies *a* (*q*) <sup>1</sup> = *ξ* (*q*) *λ* (0) = *<sup>q</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>&</sup>lt; 0. By induction, (*an*)*n*∈<sup>N</sup> is strictly negative and strictly decreasing. As stated in (P9), the function *ξ* (*q*) *λ* is strictly increasing, strictly convex and converges to −*β<sup>λ</sup>* for *x* → −∞. Thus, it hits the straight line *id*(*x*) = *x* once and only once on the negative real line at *x* (*q*) <sup>0</sup> <sup>∈</sup>] <sup>−</sup> *<sup>β</sup>λ*, 0[ (cf. (44)). This implies that the sequence *a* (*q*) *n n*∈N converges to *x* (*q*) <sup>0</sup> ∈] − *βλ*, *q* − *βλ*[. Property (P2) follows immediately. In order to prove (P3), let us fix *q* > max{0, *βλ*}, implying *a* (*q*) <sup>1</sup> = *ξ* (*q*) *λ* (0) = *q* − *β<sup>λ</sup>* > 0; notice that in this setup, the special choice *q* = 1 implies min{1,*e <sup>β</sup>λ*−1} <sup>=</sup> *e <sup>β</sup>λ*−<sup>1</sup> <sup>&</sup>lt; *<sup>q</sup>*. By induction, *a* (*q*) *n n*∈N is strictly positive and strictly increasing. Since lim*x*→<sup>∞</sup> *ξ* (*q*) *λ* (*x*) = ∞, the function *ξ* (*q*) *λ* does not necessarily hit the straight line *id*(*x*) = *x* on the positive real line. In fact, due to strict convexity (cf. (P9)), this is excluded if *ξ* (*q*)0 *λ* (0) = *q* ≥ 1. Suppose that *q* < 1. To prove that there exists a positive solution of the equation *ξ* (*q*) *λ* (*x*) = *x* it is sufficient to show that the unique global minimum of the strict convex function *h* (*q*) *λ* (*x*) := *ξ* (*q*) *λ* (*x*) − *x* is taken at some point *x*<sup>0</sup> ∈]0, ∞[ and that *h* (*q*) *λ* (*x*0) ≤ 0. It holds *h* (*q*)0 *λ* (*x*) = *q* · *e <sup>x</sup>* <sup>−</sup> 1, and therefore *<sup>h</sup>* (*q*)0 *λ* (*x*) = 0 iff *x* = *x*<sup>0</sup> = − log *q*. We have *h* (*q*) *λ* (− log *q*) = 1 − *β<sup>λ</sup>* + log *q*, which is less or equal to zero iff *q* ≤ *e βλ*−1 . It remains to show that for *q* > *β<sup>λ</sup>* and *q* > min 1 , *e βλ*−1 the sequence *a* (*q*) *n n*∈N grows faster than exponentially, i.e., there do not exist constants *c*1, *c*<sup>2</sup> ∈ R such that *a* (*q*) *<sup>n</sup>* ≤ *e c*1+*c*2*n* for all *n* ∈ N. We already know that (in the current case) *a* (*q*) *n <sup>n</sup>*→<sup>∞</sup> −→ <sup>∞</sup>. Notice that it is sufficient to verify lim sup*n*→<sup>∞</sup> log(*a* (*q*) *n*+1 ) − log(*a* (*q*) *<sup>n</sup>* ) = ∞. For the case *β<sup>λ</sup>* ≥ 0 the latter is obtained by

$$\begin{split} \left(\log\left(a\_{n+1}^{(q)}\right) - \log\left(a\_{n}^{(q)}\right)\right) &=& \log\left(\left(q-\beta\_{\lambda}\right)e^{a\_{n}^{(q)}} + \beta\_{\lambda}\left(e^{a\_{n}^{(q)}}-1\right)\right) - \log\left(qe^{a\_{n-1}^{(q)}} - \beta\_{\lambda}\right) \\ &\geq & \left(\log\left(q-\beta\_{\lambda}\right) - \log(q)\right) + \left(qe^{a\_{n-1}^{(q)}} - \beta\_{\lambda} - a\_{n-1}^{(q)}\right) \xrightarrow{a\_{n-1}^{(q)}\to\infty} \infty \end{split}$$

An analogous consideration works out for the case *β<sup>λ</sup>* < 0. Property (P4) is trivial, and (P5) to (P8) are direct implications of the already proven properties (P1) to (P4).

**Proof of Lemma 1**. (a) Let *β*<sup>A</sup> > 0, *β*<sup>H</sup> > <sup>0</sup> with *β*<sup>A</sup> 6= *β*H, *λ* ∈ R\]0, 1[, *β<sup>λ</sup>* := *λβ*<sup>A</sup> + (<sup>1</sup> − *λ*)*β*<sup>H</sup> and *q<sup>λ</sup>* := *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; max{0, *<sup>β</sup>λ*} (cf. Lemma A1). Below, we follow the lines of Linkov & Lunyova [53], appropriately adapted to our context. We have to find those *λ* ∈ R\]0, 1[ for which the following two conditions hold:


Notice that the case *q<sup>λ</sup>* = 1, *λ* ∈ R\[0, 1], cannot appear in (i), provided that (ii) holds (since due to Lemma A1 *e <sup>β</sup>λ*−<sup>1</sup> < *e <sup>q</sup>λ*−<sup>1</sup> = 1). For (i), it is easy to check that we have to require

$$\lambda \begin{cases} < \frac{\log(\beta\_{\mathcal{H}})}{\log(\beta\_{\mathcal{H}}/\beta\_{\mathcal{A}})'} & \text{if } \beta\_{\mathcal{A}} > \beta\_{\mathcal{H}} \\\\ > \frac{\log(\beta\_{\mathcal{H}})}{\log(\beta\_{\mathcal{H}}/\beta\_{\mathcal{A}})'} & \text{if } \beta\_{\mathcal{A}} < \beta\_{\mathcal{H}} . \end{cases} \tag{A1}$$

,

To proceed, straightforward analysis leads to − log(*qλ*) = arg min*x*∈<sup>R</sup> *ξ* (*qλ*) *λ* (*x*) − *x* . To check (ii), we first notice that *q<sup>λ</sup>* ≤ *e βλ*−1 iff *ξ* (*qλ*) *λ* (*x*) − *x* ≤ 0 for some *x* ∈ R. Hence, we calculate

$$
\xi\_{\lambda}^{(q\_{\lambda})} \left( -\log(q\_{\lambda}) \right) + \log(q\_{\lambda}) \le 0 \iff 1 - \lambda(\mathcal{\beta}\_{\mathcal{A}} - \mathcal{\beta}\_{\mathcal{H}}) - \mathcal{\beta}\_{\mathcal{H}} + \lambda \log \left( \frac{\mathcal{\beta}\_{\mathcal{A}}}{\mathcal{\beta}\_{\mathcal{H}}} \right) + \log(\mathcal{\beta}\_{\mathcal{H}}) \le 0
$$

$$
\iff \quad \lambda \cdot \left[ \mathcal{\beta}\_{\mathcal{H}} \left( 1 - \frac{\mathcal{\beta}\_{\mathcal{A}}}{\mathcal{\beta}\_{\mathcal{H}}} \right) + \log \left( \frac{\mathcal{\beta}\_{\mathcal{A}}}{\mathcal{\beta}\_{\mathcal{H}}} \right) \right] \le \mathcal{\beta}\_{\mathcal{H}} - 1 - \log \left( \mathcal{\beta}\_{\mathcal{H}} \right). \tag{A2}
$$

In order to isolate *λ* in (A2), one has to find out for which (*β*A, *β*H) the term in the square bracket is positive resp. zero resp. negative. To achieve this, we aim for the substitutions *x* := *β*A/*β*H, *β* = *β*<sup>H</sup> and thus study first the auxiliary function *hβ*(*x*) := log(*x*) − *β*(*x* − 1), *x* > 0, with fixed parameters *β* > 0. Straightforwardly, we obtain *h* 0 *β* (*x*) = *x* <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>β</sup>* and *<sup>h</sup>* 00 *β* (*x*) = −*x* −2 . Thus, the function *hβ*(·) is strictly concave and attains a maximum at *x* = *β* −1 . Since additionally *hβ*(1) = 0 and *h* 0 *β* (1) = 1 − *β*, there exists a second solution *z*(*β*) 6= 1 of the equation *hβ*(*x*) = 0 iff *β* 6= 1. Thus, one gets


Suppose that *λ* < 0.

**Case 1:** If *β*<sup>H</sup> = 1, then condition (ii) is not satisfied whenever *β*<sup>A</sup> 6= *β*H, since the right side of (A2) is equal to zero and the left side is strictly greater than zero. Hence, *λ*<sup>−</sup> = 0.

**Case 2:** Let *β*<sup>H</sup> > 1. If *β*<sup>A</sup> < *β*H, then condition (i) is not satisfied and hence *λ*<sup>−</sup> = 0. If *β*<sup>A</sup> > *β*H, then condition (i) is satisfied iff *λ* < ˘*λ*˘ := ˘*λ*˘(*β*A, *<sup>β</sup>*H) :<sup>=</sup> log(*β*H) log(*β*H/*β*A) <sup>&</sup>lt; 0. On the other hand, incorporating the discussion of the function *<sup>h</sup>β*(·), we see that *<sup>h</sup>β*<sup>H</sup> *β*A *β*H < 0. Thus, (A2) implies that condition (ii) is satisfied when *<sup>λ</sup>* <sup>≥</sup> *<sup>λ</sup>*˘ :<sup>=</sup> *<sup>λ</sup>*˘(*β*A, *<sup>β</sup>*H) :<sup>=</sup> *<sup>β</sup>*H−1−log(*β*H) *<sup>β</sup>*H−*β*A+log *β*A *β*H . We claim that ˘*λ*˘ <sup>&</sup>lt; *<sup>λ</sup>*˘ and conclude that the conditions (i) and (ii) are not fulfilled jointly, which leads to *λ*<sup>−</sup> = 0. To see this, we notice that due to 1 < *β*<sup>H</sup> < *β*<sup>A</sup> we get log(*β*A)/(*β*<sup>A</sup> − 1) < log(*β*H)/(*β*<sup>H</sup> − 1) and thus

$$\log(\beta\_{\mathcal{A}})(\beta\_{\mathcal{H}}-1) < \log(\beta\_{\mathcal{H}})(\beta\_{\mathcal{A}}-1)$$

$$\iff \begin{aligned} &\iff \quad \beta\_{\mathcal{H}}\log(\beta\_{\mathcal{H}}) - \beta\_{\mathcal{A}}\log(\beta\_{\mathcal{H}}) < \beta\_{\mathcal{H}}\log(\beta\_{\mathcal{H}}) - \beta\_{\mathcal{H}}\log(\beta\_{\mathcal{A}}) - \log(\beta\_{\mathcal{H}}) + \log(\beta\_{\mathcal{A}}) \\ &\iff \quad \log(\beta\_{\mathcal{H}})(\beta\_{\mathcal{H}}-\beta\_{\mathcal{A}}) + \log(\beta\_{\mathcal{H}})\log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{H}}}\right) < \log\left(\frac{\beta\_{\mathcal{H}}}{\beta\_{\mathcal{A}}}\right)(\beta\_{\mathcal{H}}-1) + \log(\beta\_{\mathcal{H}})\log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{H}}}\right) \\ &\iff \quad \frac{\log(\beta\_{\mathcal{H}})}{\log\left(\frac{\beta\_{\mathcal{H}}}{\beta\_{\mathcal{A}}}\right)} < \frac{\beta\eta\_{\mathcal{H}} - 1 - \log(\beta\_{\mathcal{H}})}{\beta\_{\mathcal{H}} - \beta\_{\mathcal{A}} + \log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{H}}}\right)} \end{aligned} \tag{A3}$$

**Case 3:** Let *<sup>β</sup>*<sup>H</sup> <sup>&</sup>lt; 1. For this, one gets *<sup>h</sup>β*<sup>H</sup> *β*A *β*H ≥ 0 for *β*<sup>A</sup> ∈]*β*H, *β*H*z*(*β*H)]. Hence, condition (ii) is satisfied if either *<sup>β</sup>*<sup>A</sup> <sup>∈</sup>]*β*H, *<sup>β</sup>*H*z*(*β*H)], or *<sup>β</sup>*<sup>A</sup> <sup>∈</sup>/]*β*H, *<sup>β</sup>*H*z*(*β*H)] and *<sup>λ</sup>* <sup>≥</sup> *<sup>λ</sup>*˘ . If *<sup>β</sup>*<sup>A</sup> <sup>&</sup>gt; *<sup>β</sup>*H*z*(*β*H), then condition (i) is trivially satisfied for all *λ* < 0. In the case *β*<sup>A</sup> < *β*H, condition (i) is satisfied whenever *λ* > ˘*λ*˘ . Notice that since <sup>0</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>A</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>H</sup> <sup>&</sup>lt; 1, an analogous consideration as in (A3) leads to ˘*λ*˘ <sup>&</sup>lt; *<sup>λ</sup>*˘ . This implies that *<sup>λ</sup>*<sup>−</sup> <sup>=</sup> *<sup>λ</sup>*˘ . The last case *<sup>β</sup>*<sup>A</sup> <sup>∈</sup>]*β*H, *<sup>β</sup>*H*z*(*β*H)] is easy to handle: since log(*β*H) log(*β*H/*β*A) <sup>&</sup>gt; 0 as well as *<sup>z</sup>β*<sup>H</sup> *β*A *β*H > 0, both conditions (i) and (ii) hold trivially.

The representation of *λ*<sup>+</sup> follows straightforwardly from the *λ*−-result and the skew symmetry (8), by employing <sup>1</sup> <sup>−</sup> *<sup>λ</sup>*˘(*β*H, *<sup>β</sup>*A) = *<sup>λ</sup>*˘(*β*A, *<sup>β</sup>*H). Alternatively, one can proceed analogously to the *λ*−-case.

Part (b) is much easier to prove: if *β*• := *β*<sup>A</sup> = *β*<sup>H</sup> > 0, then for all *λ* ∈ R\[0, 1] one gets *q<sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>=</sup> *<sup>β</sup>*• as well as *<sup>β</sup><sup>λ</sup>* <sup>=</sup> *<sup>β</sup>*•. Hence, Properties <sup>1</sup> (P2) implies that *<sup>a</sup>* (*qλ*) *<sup>n</sup>* ≡ 0 and thus it is convergent, independently of the choice *λ* ∈ R\[0, 1].

**Proof of Formula** (51)**.** For the parameter constellation in Section 3.10, we employ as upper bound for *φλ*(*x*) (*x* ∈ N0) the function

$$\overline{\phi\_{\lambda}}(\mathfrak{x}) := \begin{cases} \ \phi\_{\lambda}(0), & \text{if } \mathfrak{x} = 0, \\ 0, & \text{if } \mathfrak{x} > 0. \end{cases}$$

Notice that this method is rather crude, and gives in the other cases treated in the Sections 3.7–3.9 worse bounds than those derived there. Since *λ* ∈]0, 1[ and *α*<sup>A</sup> 6= *α*H, one has *φλ*(0) < 0. In order to derive an upper bound of the Hellinger integral, we first set *e* := 1 − *e <sup>φ</sup>λ*(0) <sup>∈</sup>]0, 1[. Hence, for all *n* ∈ N\{1} we obtain the auxiliary expression

$$\sum\_{\mathbf{x}\_{n-1}=0}^{\infty} \frac{[\varrho\_{\lambda}(\mathbf{x}\_{n-2})]^{\mathbf{x}\_{n-1}}}{\mathbf{x}\_{n-1}!} \cdot \exp\left\{\oint\_{\lambda} (\mathbf{x}\_{n-1})\right\} \leq \sum\_{\mathbf{x}\_{n-1}=0}^{\infty} \frac{[\varrho\_{\lambda}(\mathbf{x}\_{n-2})]^{\mathbf{x}\_{n-1}}}{\mathbf{x}\_{n-1}!} \cdot \exp\left\{\overleftarrow{\Phi\_{\lambda}}(\mathbf{x}\_{n-1})\right\}.$$

$$= \left[\exp\left\{\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\}-\overline{\varepsilon}\right] = \exp\left\{\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\} \cdot \left[1-\overline{\varepsilon} \cdot \exp\left\{-\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\}\right].$$

Moreover, since *β*<sup>A</sup> 6= *β*H, one gets lim*x*→<sup>∞</sup> *φλ*(*x*) = −∞ (cf. Properties 3 (P20) and Lemma A1). This–together with the nonnegativity of *ϕλ*(·)–implies

$$\sup\_{\chi \in \mathbb{N}\_0} \left\{ \exp \left\{ \phi\_\lambda(\mathfrak{x}) \right\} \cdot \left[ 1 - \overline{\mathfrak{e}} \cdot \exp \left\{ -\mathfrak{p}\_\lambda(\mathfrak{x}) \right\} \right] \right\} \\ =: \overline{\mathfrak{d}} \in ]0, 1[ \ . $$

Incorporating these considerations as well as the formulas (27) to (32), we get for *n* = 1 the relation *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*) = exp{*φλ*(*x*0)} ≤ 1 with equality iff *x*<sup>0</sup> = *x* ∗ = *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> , and–as a continuation of formula (29)– for all *n* ∈ N\{1} recall that ~*x* := (*x*0, *x*1, . . .) ∈ Ω 

*<sup>H</sup><sup>λ</sup>* (*P*A,*n*||*P*H,*n*) = ∞ ∑ *x*1=0 · · · ∞ ∑ *xn*=0 *n* ∏ *k*=1 *Z* (*λ*) *n*,*k* (~*x*) = ∞ ∑ *x*1=0 · · · ∞ ∑ *xn*−<sup>1</sup>=0 *n*−1 ∏ *k*=1 *Z* (*λ*) *n*,*k* (~*x*) · exp <sup>n</sup> (*f*A(*xn*−1)) *λ* (*f*H(*xn*−1)) (1−*λ*) <sup>−</sup> (*<sup>λ</sup> <sup>f</sup>*A(*xn*−1) + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*)*f*H(*xn*−1))<sup>o</sup> = ∞ ∑ *x*1=0 · · · ∞ ∑ *xn*−<sup>2</sup>=0 *n*−2 ∏ *k*=1 *Z* (*λ*) *n*,*k* (~*x*) · exp {−*fλ*(*xn*−2)} ∞ ∑ *xn*−<sup>1</sup>=0 [*ϕλ*(*xn*−2)] *xn*−<sup>1</sup> *xn*−<sup>1</sup> ! · exp{*φλ*(*xn*−1)} ≤ ∞ ∑ *x*1=0 · · · ∞ ∑ *xn*−<sup>2</sup>=0 *n*−2 ∏ *k*=1 *Z* (*λ*) *n*,*k* (~*x*) · exp *φλ*(*xn*−2) · h <sup>1</sup> <sup>−</sup> *<sup>e</sup>* · exp − *ϕλ*(*xn*−2) i ≤ *δ* · ∞ ∑ *x*1=0 · · · ∞ ∑ *xn*−<sup>2</sup>=0 *n*−2 ∏ *k*=1 *Z* (*λ*) *n*,*k* (~*x*) ≤ · · · ≤ *δ* b*n*/2c . (A4)

Hence, *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*) < 1 for (at least) all *n* ∈ N\{1}, and lim*n*→<sup>∞</sup> *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*) = 0.

Notice that the above proof method of formula (51) does not work for the parameter setup in Section 3.11, because there one gets *<sup>δ</sup>* <sup>=</sup> sup*x*∈N<sup>0</sup> n exp *φλ*(*x*) · h <sup>1</sup> <sup>−</sup> *<sup>e</sup>* · exp − *ϕλ*(*x*) io <sup>=</sup> 1.

**Proof of Proposition 9.** In the setup (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,4a×]0, 1[ we require *β*• := *β*<sup>A</sup> = *β*<sup>H</sup> < 1. As a linear upper bound for *φλ*(·), we employ the tangent line at *y* ≥ 0 (cf. (52))

$$\boldsymbol{\phi}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\text{tan}}(\mathbf{x}) := (p\_{\mathcal{Y}} - \boldsymbol{\mu}\_{\boldsymbol{\lambda}}) + (q\_{\mathcal{Y}} - \boldsymbol{\beta}\_{\bullet}) \cdot \mathbf{x} := (p\_{\lambda,\mathcal{Y}}^{\text{tan}} - \boldsymbol{\mu}\_{\boldsymbol{\lambda}}) + (q\_{\lambda,\mathcal{Y}}^{\text{tan}} - \boldsymbol{\beta}\_{\boldsymbol{\lambda}}) \cdot \mathbf{x} := \left(\boldsymbol{\phi}\_{\boldsymbol{\lambda}}(\boldsymbol{y}) - \boldsymbol{y} \cdot \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y})\right) + \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y}) \cdot \mathbf{x}. \tag{A5}$$

Since in the current setup PSP,4a the function *φλ*(·) is strictly increasing, the slope *φ* 0 *λ* (*y*) of the tangent line at *y* is positive. Thus we have *q<sup>y</sup>* > *β<sup>λ</sup>* and Properties 1 (P3) implies that the sequence *a* (*qy*) *n n*∈N is strictly increasing and converges to *x* (*qy*) <sup>0</sup> ∈]0, − log(*qy*)] iff *q<sup>y</sup>* ≤ min{1,*e <sup>β</sup>*•−1} <sup>=</sup> *<sup>e</sup> <sup>β</sup>*•−<sup>1</sup> < 1 (cf. (P3a)), where *x* (*qy*) 0 is the smallest solution of the equation *ξ* (*qy*) *λ* (*x*) = *q<sup>y</sup>* · *e <sup>x</sup>* <sup>−</sup> *<sup>β</sup>*• <sup>=</sup> *<sup>x</sup>*. Since *<sup>q</sup><sup>y</sup>* & *β*• for *y* → ∞ (cf. Properties 3 (P18)) and additionally *e <sup>β</sup>*•−<sup>1</sup> <sup>&</sup>gt; *<sup>β</sup>*•, there exists a large enough *<sup>y</sup>* <sup>≥</sup> <sup>0</sup> such that the sequence *a* (*qy*) *n n*∈N converges. If this *y* is also large enough to additionally guarantee *h*(*y*) < 0 for

$$h(y) := \lim\_{\mathfrak{n} \to \infty} \frac{1}{\mathfrak{n}} \log \left( \tilde{\mathcal{B}}\_{\lambda, X\_{0, \mathfrak{n}}}^{(p\_y, q\_y)} \right) \\ = \ p\_y \cdot e^{\mathbf{x}\_0^{(q\_y)}} - \mathfrak{a}\_{\lambda} \mathbf{x}$$

then one can conclude that lim*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) = 0. As a first step, for verifying *h*(*y*) < <sup>0</sup> we look for an upper bound *x* (*qy*) 0 for the fixed point *x* (*qy*) <sup>0</sup> where the latter exists for *y* ≥ *y*<sup>1</sup> (say). Notice that

$$\overline{Q}\_{\lambda}^{(q\_y)}(\mathbf{x}) := \frac{1}{2}\mathbf{x}^2 + q\_y \mathbf{x} + q\_y - \boldsymbol{\beta}\_{\bullet} \ge q\_y \cdot \boldsymbol{e}^{\mathbf{x}} - \boldsymbol{\beta}\_{\bullet} = \boldsymbol{\xi}\_{\lambda}^{(q\_y)}(\mathbf{x}) \,. \tag{A6}$$

since *Q* (*qy*) *λ* (0) = *ξ* (*qy*) *λ* (0), *Q* (*qy*)0 *λ* (0) = *ξ* (*qy*)0 *λ* (0) and *Q* (*qy*)00 *λ* (*x*) ≥ *ξ* (*qy*)00 *λ* (*x*) for *x* ∈ [0, − log(*qy*)]. For sufficiently large *y* ≥ *y*<sup>2</sup> ≥ *y*<sup>1</sup> (say), we easily obtain the smaller solution of *Q* (*qy*) *λ* (*x*) = *x* as

$$\overline{\mathbf{x}}\_{0}^{(q\_{\mathsf{f}})} = (1 - q\_{\mathsf{y}}) - \sqrt{(1 - q\_{\mathsf{y}})^2 - 2(q\_{\mathsf{y}} - \boldsymbol{\theta}\_{\mathsf{\bullet}})} = (1 - \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y}) - \boldsymbol{\beta}\_{\mathsf{\bullet}}) - \sqrt{(1 - \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y}) - \boldsymbol{\beta}\_{\mathsf{\bullet}})^2 - 2\boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y})} \geq \mathbf{x}\_{0}^{(q\_{\mathsf{f}})} \quad \text{(A7)}$$

where the expression in the root is positive since *q<sup>y</sup>* & *β*• for *y* → ∞. We now have

$$h(y) = \left. p\_y \cdot e^{\mathbf{x}\_0^{(q\_y)}} - \mathfrak{a}\_{\lambda} \right| \le \left. p\_y \cdot e^{\overline{\mathbf{x}}\_0^{(q\_y)}} - \mathfrak{a}\_{\lambda} \right| =: \overline{h}(y), \qquad \forall \, y \ge y\_2. \tag{A8}$$

Hence, it suffices to show that *h*(*y*) < 0 for some *y* ≥ *y*2. We recall from Properties 3 (P15), (P17) and (P19) that

$$\begin{array}{rclcl}\phi\_{\lambda}(\boldsymbol{y})&=&\left(\boldsymbol{a}\_{\mathcal{A}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}\right)^{\lambda}\left(\boldsymbol{a}\_{\mathcal{H}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}\right)^{1-\lambda}-\lambda\left(\boldsymbol{a}\_{\mathcal{A}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}\right)-\left(1-\lambda\right)\left(\boldsymbol{a}\_{\mathcal{H}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}\right)<&0,\\\phi\_{\lambda}^{\prime}(\boldsymbol{y})&=&\lambda\cdot\boldsymbol{\beta}\_{\mathsf{b}}\cdot\left(\frac{\boldsymbol{a}\_{\mathcal{A}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}}{\boldsymbol{a}\_{\mathcal{H}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}}\right)^{\lambda-1}+\left(1-\lambda\right)\cdot\boldsymbol{\beta}\_{\mathsf{b}}\cdot\left(\frac{\boldsymbol{a}\_{\mathcal{A}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}}{\boldsymbol{a}\_{\mathcal{H}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}}\right)^{\lambda}-\boldsymbol{\beta}\_{\mathsf{b}}>0&\text{ and that}\\\h\_{\lambda}^{\prime}\cdot\boldsymbol{y}&=&\left(\boldsymbol{a}\_{\mathcal{A}}+\boldsymbol{\beta}\_{\mathsf{b}}\cdot\boldsymbol{y}\right)^{\lambda}\cdot\lambda(1-\lambda)\cdot\boldsymbol{\beta}\_{\mathsf{b}}^{2}\cdot\left(\boldsymbol{a}\_{\mathcal{A}}-\boldsymbol{a}\_{\mathcal{H}}\right)^{2}\end{array}$$

$$\phi\_{\lambda}^{\prime\prime}(y) = -\left(\frac{\mathfrak{a}\_{\mathcal{A}} + \mathfrak{f}\_{\bullet} \cdot y}{\mathfrak{a}\_{\mathcal{H}} + \mathfrak{f}\_{\bullet} \cdot y}\right)^{\wedge} \cdot \frac{\lambda (1 - \lambda) \cdot \mathfrak{f}\_{\bullet}^{2} \cdot (\mathfrak{a}\_{\mathcal{A}} - \mathfrak{a}\_{\mathcal{H}})^{2}}{(\mathfrak{a}\_{\mathcal{A}} + \mathfrak{f}\_{\bullet} \cdot y)^{2} (\mathfrak{a}\_{\mathcal{H}} + \mathfrak{f}\_{\bullet} \cdot y)} < 0,\tag{A9}$$

which immediately implies lim*y*→<sup>∞</sup> *φλ*(*y*) = lim*y*→<sup>∞</sup> *φ* 0 *λ* (*y*) = lim*y*→<sup>∞</sup> *φ* 00 *λ* (*y*) = 0 and with l'Hospital's rule

$$\lim\_{y \to \infty} y \cdot \phi\_{\mathbb{A}}(y) = \lim\_{y \to \infty} -y^2 \cdot \phi\_{\mathbb{A}}'(y) = \lim\_{y \to \infty} \frac{y^3}{2} \cdot \phi\_{\mathbb{A}}''(y) \tag{A10}$$

$$= -\frac{1}{2} \lim\_{y \to \infty} \left( \frac{a\_{\mathcal{A}} + \beta\_{\bullet} \cdot y}{a\_{\mathcal{H}} + \beta\_{\bullet} \cdot y} \right)^{\lambda} \cdot \frac{\lambda (1 - \lambda) \cdot \beta\_{\bullet}^2 \cdot (a\_{\mathcal{A}} - a\_{\mathcal{H}})^2}{(a\_{\mathcal{A}}/y + \beta\_{\bullet})^2 (a\_{\mathcal{H}}/y + \beta\_{\bullet})} = -\frac{1}{2} \lambda (1 - \lambda) \cdot \frac{(a\_{\mathcal{A}} - a\_{\mathcal{H}})^2}{\beta\_{\bullet}} \cdot \phi\_{\mathbb{A}}$$

The formulas (A5), (A7) and (A9) imply the limits lim*y*→<sup>∞</sup> *p<sup>y</sup>* = *αλ*, lim*y*→<sup>∞</sup> *q<sup>y</sup>* = *β*•, lim*y*→<sup>∞</sup> *x* (*qy*) <sup>0</sup> = 0. Notice that *p<sup>y</sup>* < *α<sup>λ</sup>* holds trivially for all *y* ≥ 0 since the intercept (*py*−*αλ*) of the tangent line *φ* tan *λ*,*y* (·) is negative. Incorporating (A8) we therefore obtain lim*y*→<sup>∞</sup> *h*(*y*) ≤ lim*y*→<sup>∞</sup> *h*(*y*) = 0. As mentioned before, for the proof it is sufficient to show that *h*(*y*) < 0 for some *y* ≥ *y*2. This holds true if lim*y*→<sup>∞</sup> *y* · *h*(*y*) < 0. To verify this, notice first that from (A5), (A7) and (A8) we get

$$\overline{\mathcal{H}}'(y) = -p\_{\mathcal{Y}} \cdot e^{\overline{\chi}\_0^{(q\_\flat)}} \cdot \phi\_\lambda''(y) \cdot \left[1 - \frac{2 - \phi\_\lambda'(y) - \mathcal{G}\_\bullet}{\sqrt{(1 - q\_\flat)^2 - 2(q\_\flat - \mathcal{G}\_\bullet)}}\right] \\ - y \cdot \phi\_\lambda''(y) \cdot e^{\overline{\chi}\_0^{(q\_\flat)}} \stackrel{y \to \infty}{\longrightarrow} 0. \tag{A11}$$

Finally we obtain with (A10)

lim *y*→∞ *y* · *h*(*y*) = − lim *y*→∞ *y* 2 · *h* 0 (*y*) = lim *y*→∞ *p<sup>y</sup>* · *e x* (*qy*) <sup>0</sup> · *y* 2 · *φ* 00 *λ* (*y*) · 1 − 2 − *φ* 0 *λ* (*y*) <sup>−</sup> *<sup>β</sup>*• <sup>q</sup> (1 − *qy*) <sup>2</sup> − <sup>2</sup>(*q<sup>y</sup>* − *<sup>β</sup>*•) + *y* 3 · *φ* 00 *λ* (*y*) · *e x* (*qy*) 0 = 0 − *λ*(1 − *λ*) · (*α*<sup>A</sup> − *α*H) 2 *β*• < 0 .

**Proof of Corollary 1.** Part (a) follows directly from Proposition 1 (a),(b) and the limit lim*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) = 0 in the respective part (c) of the Propositions 7, 8, 9 as well as from (51). To prove part (b), according to (26) we have to verify lim inf*λ*%<sup>1</sup> {lim inf*n*→<sup>∞</sup> *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*)} = 1. From part (c) of Proposition 2 we see that this is satisfied iff lim*λ*↑<sup>1</sup> *x* (*q E λ* ) <sup>0</sup> = 0. Recall that for fixed *λ* ∈]0, 1[ we have *β<sup>λ</sup>* = *λβ*<sup>A</sup> + (<sup>1</sup> − *λ*)*β*<sup>H</sup> > 0, *q E <sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>lt; *<sup>β</sup><sup>λ</sup>* (cf. Lemma A1) and from Properties 1 (P1) the unique negative solution *x* (*q E λ* ) <sup>0</sup> ∈] − *βλ*, *q E <sup>λ</sup>* − *βλ*[ of *ξ* (*q E λ* ) *λ* (*x*) = *q E λ e <sup>x</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>=</sup> *<sup>x</sup>* (cf. (44)). Due to the continuity and boundedness of the map *λ* 7→ *x* (*q E λ* ) 0 (for *λ* ∈ [0, 1]) one gets that lim*λ*%<sup>1</sup> *x* (*q E λ* ) 0 exists and is the smallest nonpositive solution of *β*A*e <sup>x</sup>* <sup>−</sup> *<sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>x</sup>*. From this, the part (b) as well as the non-contiguity in part (c) follow immediately. The other part of (c) is a direct consequence of

Proposition 1 (a),(b) and Proposition 2 (c).

**Proof of Formula** (59) . One can proceed similarly to the proof of formula (51) above. Recall *Hλ*(*P*A,1||*P*H,1) = exp{*φλ*(*X*0)} > 1 for *X*<sup>0</sup> ∈ N cf. (28), Lemma A1 and *f*A(*X*0) 6= *f*H(*X*0) for all *X*<sup>0</sup> ∈ N . For (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,2 × (R\[0, 1]) one gets *φλ*(0) = 0, *φλ*(1) > 0, and we define for *x* ≥ 0

$$\underline{\phi\_{\Lambda}}(\mathfrak{x}) := \begin{cases} \ \phi\_{\Lambda}(1) \ , & \text{if } \mathfrak{x} = 1 \\ \ \mathbf{0} , & \text{if } \mathfrak{x} \neq 1 . \end{cases}$$

By means of the choice *e* := *ϕλ*(1) · *e <sup>φ</sup>λ*(1) <sup>−</sup> <sup>1</sup> > 0, we obtain for all *n* ∈ N\{1}

$$\sum\_{\mathbf{x}\_{n-1}=0}^{\infty} \frac{[\varrho\_{\lambda}(\mathbf{x}\_{n-2})]^{\mathbf{x}\_{n-1}}}{\mathbf{x}\_{n-1}!} \cdot \exp\left\{\underline{\phi}\_{\lambda}(\mathbf{x}\_{n-1})\right\} \geq \sum\_{\mathbf{x}\_{n-1}=0}^{\infty} \frac{[\varrho\_{\lambda}(\mathbf{x}\_{n-2})]^{\mathbf{x}\_{n-1}}}{\mathbf{x}\_{n-1}!} \cdot \exp\left\{\underline{\phi}\_{\lambda}(\mathbf{x}\_{n-1})\right\}.$$

$$= \exp\left\{\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\} + \underline{\varepsilon} = \exp\left\{\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\} \cdot \left[1 + \underline{\varepsilon} \cdot \exp\left\{-\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\}\right].$$

Incorporating

$$\inf\_{\chi \in \mathbb{N}\_0} \left\{ \exp \left\{ \phi\_\lambda(\mathfrak{x}) \right\} \cdot \left[ 1 + \underline{\mathfrak{e}} \cdot \exp \left\{ -\phi\_\lambda(\mathfrak{x}) \right\} \right] \right\} =: \underline{\delta} > 1 \,, \ \underline{\delta}$$

one can show analogously to (A4) that

$$H\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) \geq \dots \geq \underline{\delta}^{\lfloor n/2 \rfloor} \stackrel{n \to \infty}{\longrightarrow} \infty. \tag{7}$$

**Proof of the Formulas** (61)**,** (63) **and** (64)**.** In the following, we slightly adapt the above-mentioned proof of formula (59). Let us define

$$\underline{\phi\_{\lambda}}(\mathbf{x}) := \begin{cases} \ \phi\_{\lambda}(\mathbf{0}) \ , & \text{if } \mathbf{x} = \mathbf{0}, \\ \ \mathbf{0} \ , & \text{if } \mathbf{x} > \mathbf{0}. \end{cases}$$

In all respective subcases one clearly has *φλ*(0) = *φλ*(0) > 0. With *e* := *e <sup>φ</sup>λ*(0) <sup>−</sup> <sup>1</sup> <sup>&</sup>gt; <sup>0</sup> we obtain for all *n* ∈ N\{1}

$$\sum\_{\mathbf{x}\_{n-1}=0}^{\infty} \frac{[\varrho\_{\lambda}(\mathbf{x}\_{n-2})]^{\mathbf{x}\_{n-1}}}{\mathbf{x}\_{n-1}!} \cdot \exp\left\{\oint\_{\lambda} \rho\_{\lambda}(\mathbf{x}\_{n-1})\right\} \geq \sum\_{\mathbf{x}\_{n-1}=0}^{\infty} \frac{[\varrho\_{\lambda}(\mathbf{x}\_{n-2})]^{\mathbf{x}\_{n-1}}}{\mathbf{x}\_{n-1}!} \cdot \exp\left\{\underline{\phi}\_{\lambda}(\mathbf{x}\_{n-1})\right\}.$$

$$\mathcal{E} = \exp\left\{\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\} + \underline{\varepsilon} = \exp\left\{\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\} \cdot \left[1 + \underline{\varepsilon} \cdot \exp\left\{-\varrho\_{\lambda}(\mathbf{x}\_{n-2})\right\}\right].$$

By employing

$$\inf\_{\mathbf{x}\in\mathbb{N}\_{0}}\left\{\exp\left\{\phi\_{\lambda}(\mathbf{x})\right\}\cdot\left[1+\underline{\mathfrak{e}}\cdot\exp\left\{-\phi\_{\lambda}(\mathbf{x})\right\}\right]\right\} =: \underline{\delta}>1,\tag{A12}$$

one can show analogously to (A4) that

$$H\_{\lambda} \left( P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n} \right) \geq \dots \geq \underbrace{\xi^{\lfloor n/2 \rfloor}} \overset{n \to \infty}{\longrightarrow} \infty.$$

Notice that this method does not work for the parameter cases PSP,4a ∪ PSP,4b, since there the infimum in (A12) is equal to one.

**Proof of Proposition 13.** In the setup (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,4a × (R\[0, 1]) we require *β*• := *β*<sup>A</sup> = *β*<sup>H</sup> < 1. As in the proof of Proposition 9, we stick to the tangent line *φ* tan *λ*,*y* (·) at *y* ≥ 0 (cf. (52)) as a linear lower bound for *φλ*(·), i.e., we use the function

$$\boldsymbol{\Phi}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\text{tan}}(\mathbf{x}) := \left(p\_{\mathcal{Y}} - \boldsymbol{a}\_{\boldsymbol{\lambda}}\right) + \left(q\_{\mathcal{Y}} - \boldsymbol{\beta}\_{\bullet}\right) \cdot \mathbf{x} := \left(p\_{\lambda,\mathcal{Y}}^{\text{tan}} - \boldsymbol{a}\_{\boldsymbol{\lambda}}\right) + \left(q\_{\lambda,\mathcal{Y}}^{\text{tan}} - \boldsymbol{\beta}\_{\boldsymbol{\lambda}}\right) \cdot \mathbf{x} := \left(\boldsymbol{\phi}\_{\boldsymbol{\lambda}}(\boldsymbol{y}) - \boldsymbol{y} \cdot \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y})\right) + \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y}) \cdot \mathbf{x} . \tag{A13}$$

As already mentioned in Section 3.21, on PSP,4a the function *φλ*(·) is strictly decreasing and converges to 0. Thus, for all *y* ≥ 0 the slope *φ* 0 *λ* (*y*) of the tangent line at *y* is negative, which implies that *q<sup>y</sup>* < *β<sup>λ</sup>* = *β*•. For *λ* ∈ R\[0, 1] there clearly may hold *q<sup>y</sup>* < 0 for some *y* ∈ R. However, there exists a sufficiently large *y*<sup>1</sup> > 0 such that *q<sup>y</sup>* > 0 for all *y* > *y*1, since lim*y*→<sup>∞</sup> *φ* 0 *λ* (*y*) = 0 and hence *q<sup>y</sup>* % *β*• > 0 for *<sup>y</sup>* <sup>→</sup> <sup>∞</sup>. Thus, let us suppose that *<sup>y</sup>* <sup>&</sup>gt; *<sup>y</sup>*1. Then, the sequence *a* (*qy*) *n n*∈N is strictly negative, strictly decreasing and converges to *x* (*qy*) <sup>0</sup> ∈] − *β*•, *q<sup>y</sup>* − *β*•[ (cf. Properties 1 (P1)). If there is some *y* ≥ *y*<sup>1</sup> such that *h*(*y*) > 0 with

$$h(y) := \lim\_{\mathfrak{n}\to\infty} \frac{1}{\mathfrak{n}} \log \left( \widetilde{B}^{(p\_{\mathfrak{Y}}, q\_{\mathfrak{Y}})}\_{\lambda, X\_0, \mathfrak{n}} \right) \\ = \ p\_{\mathcal{Y}} \cdot e^{\mathbf{x}\_0^{(q\_{\mathfrak{Y}})}} - \mathfrak{a}\_{\lambda} \lambda$$

then one can conclude that lim*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) = ∞. Let us at first consider the case *α<sup>λ</sup>* ≥ 0. By employing *p<sup>y</sup>* & *α<sup>λ</sup>* for *y* → ∞, one gets *p<sup>y</sup>* > 0 for all *y* ≥ 0. Analogously to the proof of Proposition 9, we now look for a lower bound *x* (*qy*) 0 of the fixed point *x* (*qy*) 0 . Notice that *x* (*qy*) <sup>0</sup> > −*β*• implies

$$\underline{\mathbf{Q}}\_{\lambda}^{(q\_{\mathcal{Y}})}(\mathbf{x}) := \frac{e^{-\beta \mathbf{\cdot}}}{2} \cdot q\_{\mathcal{Y}} \cdot \mathbf{x}^2 + q\_{\mathcal{Y}} \cdot \mathbf{x} + q\_{\mathcal{Y}} - \beta \mathbf{\cdot} \le q\_{\mathcal{Y}} \cdot e^{\mathbf{x}} - \beta \mathbf{\cdot} = \boldsymbol{\xi}\_{\lambda}^{(q\_{\mathcal{Y}})}(\mathbf{x}) \,, \tag{A14}$$

since *Q* (*qy*) *λ* (0) = *ξ* (*qy*) *λ* (0) < 0, *Q* (*qy*)0 *λ* (0) = *ξ* (*qy*)0 *λ* (0) > 0 and 0 < *Q* (*qy*)00 *λ* (*x*) < *ξ* (*qy*)00 *λ* (*x*) for *x* ∈] − *β*•, 0]. Thus, the negative solution *x* (*qy*) 0 of the equation *Q* (*qy*) *λ* (*x*) = *x* (which definitely exists) implies that there holds *x* (*qy*) <sup>0</sup> ≤ *x* (*qy*) 0 . We easily obtain

$$\begin{split} \underline{\mathbf{x}}\_{0}^{(q\_{\mathsf{f}})} &= \quad \frac{e^{\mathsf{f}\bullet}}{q\_{\mathsf{f}}} \left[ (1 - q\_{\mathsf{f}}) - \sqrt{(1 - q\_{\mathsf{f}})^{2} - 2e^{-\mathsf{f}\bullet}q\_{\mathsf{f}}(q\_{\mathsf{f}} - \mathsf{f}\bullet)} \right] \\ &= \quad \frac{e^{\mathsf{f}\bullet}}{\underline{\boldsymbol{\theta}}\_{\lambda}^{\prime}(\boldsymbol{y}) + \boldsymbol{\beta}\_{\mathsf{\bullet}}} \left[ (1 - \underline{\boldsymbol{\phi}}\_{\lambda}^{\prime}(\boldsymbol{y}) - \boldsymbol{\beta}\_{\mathsf{\bullet}}) - \sqrt{(1 - \boldsymbol{\phi}\_{\lambda}^{\prime}(\boldsymbol{y}) - \boldsymbol{\beta}\_{\mathsf{\bullet}})^{2} - 2 \cdot e^{-\mathsf{f}\bullet}q\_{\mathsf{f}} \cdot \boldsymbol{\phi}\_{\lambda}^{\prime}(\boldsymbol{y})} \right] < 0. \end{split} \tag{A15}$$

Since

$$h(y) = \left. p\_y \cdot e^{\mathbf{x}\_0^{(q\_y)}} - \mathfrak{a}\_{\lambda} \ge \left. p\_y \cdot e^{\mathbf{x}\_0^{(q\_y)}} - \mathfrak{a}\_{\lambda} \right. =: \underline{h}(y) \,, \tag{A16}$$

it is sufficient to show *h*(*y*) > 0 for some *y* > *y*1. We recall from Properties 3 (P15), (P17) and (P19) that

$$\begin{array}{rclcrcl}\phi\_{\lambda}(y) &=& \left(a\_{\mathcal{A}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}\right)^{\lambda} \left(a\_{\mathcal{H}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}\right)^{1-\lambda} - \lambda \left(a\_{\mathcal{A}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}\right) - (1-\lambda) \left(a\_{\mathcal{H}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}\right) &> 0, \\ \phi\_{\lambda}^{\prime}(y) &=& \lambda \cdot \boldsymbol{\beta}\_{\bullet} \cdot \left(\frac{a\_{\mathcal{A}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}}{a\_{\mathcal{H}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}}\right)^{\lambda-1} + (1-\lambda) \cdot \boldsymbol{\beta}\_{\bullet} \cdot \left(\frac{a\_{\mathcal{A}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}}{a\_{\mathcal{H}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}}\right)^{\lambda} - \boldsymbol{\beta}\_{\bullet} \cdot < 0 &\quad \text{and} \\ \phi\_{\lambda}^{\prime\prime}(y) &=& -\left(\frac{a\_{\mathcal{A}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}}{a\_{\mathcal{H}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y}}\right)^{\lambda} \cdot \frac{\lambda(1-\lambda) \cdot \boldsymbol{\beta}\_{\bullet}^{2} \cdot (a\_{\mathcal{A}} - a\_{\mathcal{H}})^{2}}{(a\_{\mathcal{A}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y})^{2}(a\_{\mathcal{H}} + \boldsymbol{\beta}\_{\bullet} \cdot \boldsymbol{y})} > 0,\end{array}$$

which immediately implies lim*y*→<sup>∞</sup> *φλ*(*y*) = lim*y*→<sup>∞</sup> *φ* 0 *λ* (*y*) = lim*y*→<sup>∞</sup> *φ* 00 *λ* (*y*) = 0, and by means of l'Hospital's rule

$$\lim\_{y \to \infty} y \cdot \phi\_{\lambda}(y) = \lim\_{y \to \infty} -y^2 \cdot \phi\_{\lambda}'(y) = \lim\_{y \to \infty} \frac{y^3}{2} \cdot \phi\_{\lambda}''(y) \tag{A18}$$

$$= -\frac{1}{2} \lim\_{y \to \infty} \left(\frac{\mathfrak{a}\_{\mathcal{A}} + \mathfrak{f}\_{\mathsf{\bullet}} \cdot y}{\mathfrak{a}\_{\mathcal{H}} + \mathfrak{f}\_{\mathsf{\bullet}} \cdot y}\right)^{\lambda} \cdot \frac{\lambda (1 - \lambda) \cdot \mathfrak{f}\_{\mathsf{\bullet}}^2 \cdot (\mathfrak{a}\_{\mathcal{A}} - \mathfrak{a}\_{\mathcal{H}})^2}{(\mathfrak{a}\_{\mathcal{A}}/y + \mathfrak{f}\_{\mathsf{\bullet}})^2 (\mathfrak{a}\_{\mathcal{H}}/y + \mathfrak{f}\_{\mathsf{\bullet}})} = -\frac{1}{2} \lambda (1 - \lambda) \cdot \frac{(\mathfrak{a}\_{\mathcal{A}} - \mathfrak{a}\_{\mathcal{H}})^2}{\mathfrak{f}\_{\mathsf{\bullet}}} \cdot \mathfrak{a}\_{\mathcal{H}}$$

The Formulas (A13), (A15), (A17) imply the limits lim*y*→<sup>∞</sup> *p<sup>y</sup>* = *αλ*, lim*y*→<sup>∞</sup> *q<sup>y</sup>* = *β*• and lim*y*→<sup>∞</sup> *x* (*qy*) <sup>0</sup> = 0 iff *β*• ≤ 1. The latter is due to the fact that for *β*• > 1 one gets with (A15) lim*y*→<sup>∞</sup> *x* (*qy*) <sup>0</sup> = *<sup>e</sup> β*• *β*• - (1 − *β*•) − p (1 − *β*•) 2 = *<sup>e</sup> β*• *β*• - 2 − 2*β*• 6= 0. In the following, let us assume *β*• < 1 (the reason why we exclude the case *β*• = 1 is explained below). One gets lim*y*→<sup>∞</sup> *h*(*y*) ≥ lim*y*→<sup>∞</sup> *h*(*y*) = 0. Since we have to prove that *h*(*y*) > 0 for some *y* > *y*1, it is sufficient to show that lim*y*→<sup>∞</sup> *y* · *h*(*y*) > 0. To verify the latter, we first derive with l'Hospital's rule and with (A17), (A18)

$$\begin{split} & \lim\_{y \to \infty} y \cdot \left( 1 - e^{\Phi\_{\mathsf{A}}^{(\mathsf{q})}} \right) = \lim\_{y \to \infty} y^2 \cdot e^{\Phi\_{\mathsf{A}}^{(\mathsf{q})}} \cdot \left( \frac{\partial}{\partial y} \Delta\_{\mathsf{Q}}^{(\mathsf{q})} \right) \\ & = \lim\_{y \to \infty} \left\{ y^2 \cdot \frac{-e^{\mathsf{F}\mathsf{A}} \cdot \Phi\_{\mathsf{A}}^{\prime\prime}(y)}{\left( \phi\_{\mathsf{A}}^{\prime}(y) + \beta\_{\mathsf{A}} \right)^2} \cdot \left[ (1 - q\_{\mathsf{Y}}) - \sqrt{(1 - q\_{\mathsf{Y}})^2 - 2e^{-\mathsf{F}\mathsf{A}} q\_{\mathsf{Y}} (q\_{\mathsf{Y}} - \beta\_{\mathsf{A}} \mathsf{A})} \right] \\ & \qquad \qquad + \frac{e^{\mathsf{F}\mathsf{A}}}{q\_{\mathsf{Y}}} \cdot \left[ -y^2 \cdot \phi\_{\mathsf{A}}^{\prime\prime}(y) - \frac{-2y^2 \phi\_{\mathsf{A}}^{\prime\prime}(y)(1 - q\_{\mathsf{Y}}) - 2y^2 \phi\_{\mathsf{A}}^{\prime\prime}(y) e^{-\beta\_{\mathsf{A}}} q\_{\mathsf{Y}} - 2y^2 \phi\_{\mathsf{A}}^{\prime\prime}(y) e^{-\beta\_{\mathsf{Y}}} \phi\_{\mathsf{A}}^{\prime}(y)}{2 \cdot \sqrt{(1 - q\_{\mathsf{Y}})^2 - 2e^{-\mathsf{F}\mathsf{A}} q\_{\mathsf{Y}} (q\_{\mathsf{Y}} - \beta\_{\mathsf{Y}})}} \right] \right\} \\ &= \text{ } = \text{ } \tag{A19} \end{split} \tag{A10} \tag{A10}$$

Notice that without further examination this limit would not necessarily hold for *β*• = 1, since then the denominator in (A19) converges to zero. With (A13), (A16), (A18) and (A19) we finally obtain

$$\lim\_{y \to \infty} y \cdot \underline{h}(y) \quad = \lim\_{y \to \infty} \left\{ \left( y \cdot \phi\_{\lambda}(y) - y^2 \cdot \phi\_{\lambda}'(y) \right) \cdot e^{\underline{\mathbf{a}}\_0^{(q\_y)}} - y \cdot \left( 1 - e^{\underline{\mathbf{a}}\_0^{(q\_y)}} \right) \mathbf{a}\_{\lambda} \right\}$$

$$= -\lambda (1 - \lambda) \frac{(\mathbf{a}\_{\mathcal{A}} - \mathbf{a}\_{\mathcal{H}})^2}{\beta\_{\mathsf{\bullet}}} \;>\; 0. \tag{A20}$$

Let us now consider the case *α<sup>λ</sup>* < 0. The proof works out almost completely analogous to the case *α<sup>λ</sup>* ≥ 0. We indicate the main differences. Since *p<sup>y</sup>* & *α<sup>λ</sup>* < 0 and *q<sup>y</sup>* % *β*• ∈]0, 1[ for *y* → ∞, there is a sufficiently large *y*<sup>2</sup> > *y*1, such that *p<sup>y</sup>* < 0 and *q<sup>y</sup>* > 0. Thus,

$$\overline{\mathbb{Q}}\_{\lambda}^{(q\_{\mathcal{Y}})}(\mathbf{x}) := \frac{q\_{\mathcal{Y}}}{2} \cdot \mathbf{x}^2 + q\_{\mathcal{Y}} \cdot \mathbf{x} + q\_{\mathcal{Y}} - \mathcal{\mathfrak{H}} \ge \mathfrak{f}\_{\lambda}^{(q\_{\mathcal{Y}})}(\mathbf{x}) = q\_{\mathcal{Y}} e^{\mathbf{x}} - \mathcal{\mathfrak{H}} \qquad \text{for } \mathbf{x} \in ]-\infty, \mathbf{0}[.]$$

The corresponding (existing) smaller solution of *Q* (*qy*) *λ* (*x*) = *x* is

$$\overline{\mathfrak{X}}\_{0}^{(q\_{\mathcal{Y}})} = \frac{1}{q\_{\mathcal{Y}}} \left[ (1 - q\_{\mathcal{Y}}) - \sqrt{(1 - q\_{\mathcal{Y}})^2 - 2q\_{\mathcal{Y}}(q\_{\mathcal{Y}} - \beta\_{\bullet})} \right] \,,$$

having the same form as the solution (A15) with *e* <sup>−</sup>*β*• substituted by 1. Notice that there clearly holds *x* (*qy*) <sup>0</sup> < *x* (*qy*) <sup>0</sup> < 0. However, since *p<sup>y</sup>* < 0, we now get *h*(*y*) = *p<sup>y</sup>* · *e x* (*qy*) <sup>0</sup> − *α<sup>λ</sup>* ≥ *p<sup>y</sup>* · *e x* (*qy*) <sup>0</sup> − *α<sup>λ</sup>* =: *h*(*y*), as in (A16). Since all calculations (A17) to (A20) remain valid (with *e* <sup>−</sup>*β*• substituted by 1), this proof is finished.

#### *Appendix A.2. Proofs and Auxiliary Lemmas for Section 5*

We start with two lemmas which will be useful for the proof of Theorem 3. They deal with the sequence *a* (*qλ*) *n n*∈N from (36).

**Lemma A2.** *For arbitrarily fixed parameter constellation* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P×]0, 1[*, suppose that q<sup>λ</sup>* > 0 *and* lim*λ*%<sup>1</sup> *q<sup>λ</sup>* = *β*<sup>A</sup> *holds. Then one gets the limit*

$$\forall \ n \in \mathbb{N}: \quad \lim\_{\lambda \nearrow 1} a\_n^{(q\_\lambda)} = 0. \tag{A21}$$

**Proof.** This can be easily seen by induction: for *n* = 1 there clearly holds

$$\lim\_{\lambda \nearrow 1} a\_1^{(q\_\lambda)} = \lim\_{\lambda \nearrow 1} (q\_\lambda - \beta\_\lambda) \, = \, \not\!\! \mathcal{A} - \beta \, \_\mathcal{A} = \, 0 \, .$$

Assume now that lim*λ*%<sup>1</sup> *a* (*qλ*) *<sup>k</sup>* = 0 holds for all *k* ∈ N, *k* ≤ *n* − 1, then

$$\lim\_{\lambda \nearrow 1} a\_n^{(q\_\lambda)} = \lim\_{\lambda \nearrow 1} (q\_\lambda \cdot e^{a\_{n-1}^{(q\_\lambda)}} - \beta\_\lambda) = \beta\_\mathcal{A} \cdot 1 - \beta\_\mathcal{A} = \begin{array}{c} \square \end{array}$$

**Lemma A3.** *In addition to the assumptions of Lemma A2, suppose that λ* 7→ *q<sup>λ</sup> is continuously differentiable on* ]0, 1[ *and that the limit l* := lim*λ*%<sup>1</sup> *∂ q<sup>λ</sup> ∂λ is finite. Then, for all n* ∈ N *one obtains*

$$\lim\_{\lambda \nearrow 1} \frac{\partial a\_n^{(q\_\lambda)}}{\partial \lambda} = \ u\_n := \begin{cases} \frac{l + \beta \mu - \beta\_\mathcal{A}}{1 - \beta\_\mathcal{A}} \cdot \left(1 - (\beta\_\mathcal{A})^n\right), & \text{if } \beta\_\mathcal{A} \ne 1, \\\\ n \cdot (l + \beta\_\mathcal{H} - 1) \,, & \text{if } \beta\_\mathcal{A} = 1, \end{cases} \tag{A22}$$

*which is the unique solution of the linear recursion equation*

$$u\_n = l + \beta \mathcal{A} - \beta \mathcal{A} + \beta \mathcal{A} \cdot u\_{n-1} \,, \qquad u\_0 = 0 \,. \tag{A23}$$

*Furthermore, for all n* ∈ N *there holds*

$$\sum\_{k=1}^{n} \lim\_{\lambda \nearrow 1} \frac{\partial a\_k^{(q\_\lambda)}}{\partial \lambda} = \sum\_{k=1}^{n} u\_k = \begin{cases} \frac{l + \beta \mathfrak{z} - \mathfrak{f}\_{\mathcal{A}}}{1 - \beta \mathcal{A}} \cdot \left[ n - \frac{\mathfrak{f}\_{\mathcal{A}}}{1 - \beta \mathcal{A}} \left( 1 - (\mathfrak{f}\_{\mathcal{A}})^n \right) \right], & \text{if } \beta \mathcal{A} \neq 1, \\\\ \frac{n \cdot (n+1)}{2} \cdot (l + \beta \mathfrak{z} - 1) \; , & \text{if } \beta \mathcal{A} = 1. \end{cases}$$

**Proof.** Clearly, *u<sup>n</sup>* defined by (A22) is the unique solution of (A23). We prove by induction that lim*λ*%<sup>1</sup> *∂ a* (*qλ* ) *n ∂λ* = *u<sup>n</sup>* holds. For *n* = 1 one gets

$$\lim\_{\lambda \nearrow 1} \frac{\partial a\_1^{(q\_\lambda)}}{\partial \lambda} = \lim\_{\lambda \nearrow 1} \frac{\partial \left(q\_\lambda - \mathcal{B}\_\lambda\right)}{\partial \lambda} = \left.l - \left(\mathcal{B}\_\mathcal{A} - \mathcal{B}\_\mathcal{H}\right)\right| = \mu\_1.$$

Suppose now that (A22) holds for all *k* ∈ N, *k* ≤ *n* − 1. Then, by incorporating (A21) we obtain

$$\lim\_{\lambda \nearrow 1} \frac{\partial a\_n^{(q\_\lambda)}}{\partial \lambda} = \lim\_{\lambda \nearrow 1} \frac{\partial}{\partial \lambda} \left( q\_\lambda \cdot e^{a\_{n-1}^{(q\_\lambda)}} - \beta\_\lambda \right) \\ = \lim\_{\lambda \nearrow 1} e^{a\_{n-1}^{(q\_\lambda)}} \cdot \left( \frac{\partial q\_\lambda}{\partial \lambda} + q\_\lambda \frac{\partial a\_{n-1}^{(q\_\lambda)}}{\partial \lambda} \right) - (\beta\_\mathcal{A} - \beta\_\mathcal{H})$$
 
$$= \quad l - (\beta\_\mathcal{A} - \beta\_\mathcal{H}) + \beta\_\mathcal{A} \cdot u\_{n-1} = u\_n.$$

The remaining assertions follow immediately.

We are now ready to give the

**Proof of Theorem** 3. (a) Recall that for the setup (*β*A, *β*H, *α*A, *α*H) ∈ (PNI ∪ PSP,1) we chose the intercept as *p<sup>λ</sup>* := *p E λ* := *α λ* A *α* 1−*λ* H and the slope as *q<sup>λ</sup>* := *q E λ* := *β λ* A *β* 1−*λ* H , which in (39) lead to the exact value *<sup>V</sup>λ*,*X*0,*<sup>n</sup>* of the Hellinger integral. Because of *<sup>p</sup><sup>λ</sup> qλ β<sup>λ</sup>* − *α<sup>λ</sup>* = <sup>0</sup> as well as lim*λ*%<sup>1</sup> *q<sup>λ</sup>* = *β*A, we obtain by using (38) and Lemma A2 for all *X*<sup>0</sup> ∈ N and for all *n* ∈ N

$$\lim\_{\lambda \nearrow 1} V\_{\lambda, \mathbf{X}\_0 \mu} := \lim\_{\lambda \nearrow 1} \exp \left\{ a\_n^{(q\_\lambda)} \cdot \mathbf{X}\_0 + \sum\_{k=1}^n b\_k^{(p\_\lambda, q\_\lambda)} \right\} \\ = \lim\_{\lambda \nearrow 1} \exp \left\{ a\_n^{(q\_\lambda)} \cdot \mathbf{X}\_0 + \frac{\mathfrak{a}\_\mathcal{A}}{\mathfrak{f}\_\mathcal{A}} \sum\_{k=1}^n a\_k^{(q\_\lambda)} \right\} \\ = 1\_\mathcal{M}$$

which leads by (68) to

$$\begin{split} I(P\_{\mathcal{A},\boldsymbol{n}}||P\_{\mathcal{H},\boldsymbol{n}}) &= \lim\_{\lambda \nearrow 1} \frac{1 - H\_{\lambda}(P\_{\mathcal{A},\boldsymbol{n}}||P\_{\mathcal{H},\boldsymbol{n}})}{\lambda \cdot (1 - \lambda)} = \lim\_{\lambda \nearrow 1} \frac{1 - V\_{\lambda, \mathcal{X}\_{0,\boldsymbol{n}}}}{\lambda \cdot (1 - \lambda)} \\ &= \lim\_{\lambda \nearrow 1} \frac{-V\_{\lambda, \mathcal{X}\_{0,\boldsymbol{n}}}}{1 - 2\lambda} \cdot \frac{\partial}{\partial \lambda} \left[ a\_{\boldsymbol{n}}^{(q\_{\boldsymbol{\lambda}})} \cdot X\_{0} + \frac{p\_{\boldsymbol{\lambda}}}{q\_{\boldsymbol{\lambda}}} \sum\_{k=1}^{n} a\_{k}^{(q\_{\boldsymbol{\lambda}})} \right] \\ &= \lim\_{\lambda \nearrow 1} \left[ \frac{\partial a\_{\boldsymbol{n}}^{(q\_{\boldsymbol{\lambda}})}}{\partial \lambda} \cdot X\_{0} + \left( \frac{\partial}{\partial \lambda} \frac{p\_{\boldsymbol{\lambda}}}{q\_{\boldsymbol{\lambda}}} \right) \cdot \sum\_{k=1}^{n} a\_{k}^{(q\_{\boldsymbol{\lambda}})} + \frac{p\_{\boldsymbol{\lambda}}}{q\_{\boldsymbol{\lambda}}} \cdot \sum\_{k=1}^{n} \frac{\partial a\_{k}^{(q\_{\boldsymbol{\lambda}})}}{\partial \lambda} \right]. \end{split} \tag{A24}$$

For further analysis, we use the obvious derivatives

$$\frac{\partial \, p\_{\lambda}}{\partial \lambda} = \, p\_{\lambda} \log \left( \frac{a\_{\mathcal{A}}}{a\_{\mathcal{H}}} \right), \qquad \frac{\partial}{\partial \lambda} \frac{p\_{\lambda}}{q\_{\lambda}} = \, \frac{p\_{\lambda}}{q\_{\lambda}} \log \left( \frac{a\_{\mathcal{A}} \mathcal{\beta}\_{\mathcal{H}}}{a\_{\mathcal{H}} \mathcal{\beta}\_{\mathcal{A}}} \right), \qquad \frac{\partial q\_{\lambda}}{\partial \lambda} = \, q\_{\lambda} \log \left( \frac{\mathcal{\beta}\_{\mathcal{A}}}{\mathcal{\beta}\_{\mathcal{H}}} \right), \tag{A25}$$

where the subcase (*β*A, *β*H, *α*A, *α*H) ∈ PNI (with *p<sup>λ</sup>* ≡ 0) is consistently covered. From (A25) and Lemma A3 we deduce

$$\lim\_{\lambda \nearrow 1} \frac{\partial a\_{\boldsymbol{\eta}}^{(q\_{\boldsymbol{\lambda}})} }{\partial \lambda} \cdot \mathbf{X}\_{0} = \begin{cases} \left(\boldsymbol{\beta}\_{\mathcal{A}} \log\left(\frac{\boldsymbol{\beta}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{H}}}\right) - (\boldsymbol{\beta}\_{\mathcal{A}} - \boldsymbol{\beta}\_{\mathcal{H}})\right) \cdot \frac{1 - (\boldsymbol{\beta}\_{\mathcal{A}})^{\boldsymbol{n}}}{1 - \boldsymbol{\beta}\_{\mathcal{A}}} \cdot \mathbf{X}\_{0} & \text{if } \boldsymbol{\beta}\_{\mathcal{A}} \neq \mathbf{1}\_{\mathcal{H}}\\\ n \cdot \left(\boldsymbol{\beta}\_{\mathcal{A}} \log\left(\frac{\boldsymbol{\beta}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{H}}}\right) - (\boldsymbol{\beta}\_{\mathcal{A}} - \boldsymbol{\beta}\_{\mathcal{H}})\right) \cdot \mathbf{X}\_{0} & \text{if } \boldsymbol{\beta}\_{\mathcal{A}} = \mathbf{1}\_{\mathcal{H}} \end{cases}$$

and by means of (A21)

$$\forall n \in \mathbb{N}: \quad \lim\_{\lambda \nearrow 1} \left[ \left( \frac{\partial}{\partial \lambda} \frac{p\_{\lambda}}{q\_{\lambda}} \right) \cdot \sum\_{k=1}^{n} a\_{k}^{(q\_{\lambda})} \right] = \text{ o.}$$

For the last expression in (A24) we again apply Lemma A3 to end up with

$$\lim\_{\lambda \searrow 1} \frac{p\_{\lambda}}{q\_{\lambda} \cdot 1} \cdot \sum\_{k=1}^{n} \frac{\partial}{\partial \lambda} a\_{k}^{(q\_{\lambda})} = \begin{cases} \frac{a\_{\mathcal{A}} \cdot \left[\beta\_{\mathcal{A}} \log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{B}}}\right) - \left(\beta\_{\mathcal{A}} - \beta\_{\mathcal{H}}\right)\right]}{\beta\_{\mathcal{A}} (1 - \beta\_{\mathcal{A}})} \cdot \left[n - \frac{\beta\_{\mathcal{A}}}{1 - \beta\_{\mathcal{A}}} \left(1 - \left(\beta\_{\mathcal{A}}\right)^{n}\right)\right], & \text{if } \beta\_{\mathcal{A}} \neq 1, \\\ n \cdot (n+1) \frac{a\_{\mathcal{A}}}{2\beta\_{\mathcal{A}}} \cdot \left[\beta\_{\mathcal{A}} \log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{B}}}\right) - \left(\beta\_{\mathcal{A}} - \beta\_{\mathcal{H}}\right)\right], & \text{if } \beta\_{\mathcal{A}} = 1, \end{cases} \tag{A26}$$

which finishes the proof of part (a). To show part (b), for the corresponding setup (*β*A, *β*H, *α*A, *α*H) ∈ PSP\PSP,1 let us first choose – according to (45) in Section 3.4—the intercept as *p<sup>λ</sup>* := *p L λ* := *α λ* A *α* 1−*λ* H and the slope as *q<sup>λ</sup>* := *q L λ* := *β λ* A *β* 1−*λ* H , which in part (b) of Proposition 6 lead to the lower bounds *B L λ*,*X*0,*n* of the Hellinger integral. This is formally the same choice as in part (a) satisfying lim*λ*%<sup>1</sup> *p<sup>λ</sup>* = *α*A, lim*λ*%<sup>1</sup> *<sup>q</sup><sup>λ</sup>* <sup>=</sup> *<sup>β</sup>*<sup>A</sup> but in contrast to (a) we now have *<sup>p</sup><sup>λ</sup> qλ β<sup>λ</sup>* − *α<sup>λ</sup>* 6= 0 but nevertheless

$$\lim\_{\lambda \nearrow 1} \frac{p\_\lambda}{q\_\lambda} \,\beta\_\lambda - \alpha\_\lambda \, = \, 0.$$

From this, (38), part (b) of Proposition 6 and Lemma A2 we obtain

$$\lim\_{\lambda \nearrow 1} B^L\_{\lambda, \mathbf{X}\_0, \boldsymbol{\mu}} = \lim\_{\lambda \nearrow 1} \exp \left\{ a\_n^{(q\_\lambda)} \cdot \mathbf{X}\_0 + \frac{p\_\lambda}{q\_\lambda} \sum\_{k=1}^n a\_k^{(q\_\lambda)} + n \cdot \left( \frac{p\_\lambda}{q\_\lambda} \beta\_\lambda - a\_\lambda \right) \right\} \tag{A27}$$

and hence

$$I(P\_{\mathcal{A},\pi}||P\_{\mathcal{H},\pi}) \leq \varliminf\_{\lambda \nearrow 1} \frac{1 - B^L\_{\lambda, X\_0, \mu}}{\lambda \cdot (1 - \lambda)} = \varlimsup\_{\lambda \nearrow 1} \frac{-B^L\_{\lambda, X\_0, \mu}}{1 - 2\lambda} \cdot \frac{\partial}{\partial \lambda} \left[ a^{(q\_\lambda)}\_{\boldsymbol{\mu}} X\_0 + \frac{p\_\lambda}{q\_\lambda} \sum\_{k=1}^n a^{(q\_\lambda)}\_k + n \left( \frac{p\_\lambda}{q\_\lambda} \beta\_\lambda - a\_\lambda \right) \right]$$

$$= \varliminf\_{\lambda \nearrow 1} \left[ \frac{\partial a^{(q\_\lambda)}\_\eta}{\partial \lambda} X\_0 + \left( \frac{\partial}{\partial \lambda} \frac{p\_\lambda}{q\_\lambda} \right) \sum\_{k=1}^n a^{(q\_\lambda)}\_k + \frac{p\_\lambda}{q\_\lambda} \sum\_{k=1}^n \frac{\partial a^{(q\_\lambda)}\_k}{\partial \lambda} + n \frac{\partial}{\partial \lambda} \left( \frac{p\_\lambda}{q\_\lambda} \beta\_\lambda - a\_\lambda \right) \right]. \tag{A28}$$

In the current setup, the first three expressions in (A28) can be evaluated in exactly the same way as in (A25) to (A26), and for the last expression one has the limit

$$\begin{split} \frac{\partial}{\partial\lambda} \left( \frac{p\_{\lambda}}{q\_{\lambda}} \boldsymbol{\beta}\_{\lambda} - \boldsymbol{a}\_{\lambda} \right) &= \quad \frac{p\_{\lambda}}{q\_{\lambda}} \log \left( \frac{\boldsymbol{a}\_{\mathcal{A}} \boldsymbol{\beta}\_{\mathcal{H}}}{\boldsymbol{a}\_{\mathcal{H}} \boldsymbol{\beta}\_{\mathcal{A}}} \right) \cdot \boldsymbol{\beta}\_{\lambda} + \, \frac{p\_{\lambda}}{q\_{\lambda}} \cdot \left( \boldsymbol{\beta}\_{\mathcal{A}} - \boldsymbol{\beta}\_{\mathcal{H}} \right) - \left( \boldsymbol{a}\_{\mathcal{A}} - \boldsymbol{a}\_{\mathcal{H}} \right) \\ \stackrel{\lambda \nearrow 1}{\longrightarrow} \quad \boldsymbol{a}\_{\mathcal{A}} \left[ \log \left( \frac{\boldsymbol{a}\_{\mathcal{A}} \boldsymbol{\beta}\_{\mathcal{H}}}{\boldsymbol{a}\_{\mathcal{H}} \boldsymbol{\beta}\_{\mathcal{A}}} \right) - \frac{\boldsymbol{\beta}\_{\mathcal{H}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \right] + \boldsymbol{a}\_{\mathcal{H}} \end{split}$$

which finishes the proof of part (b).

**Proof of Theorem** 4. Let us fix (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP\PSP,1, *<sup>X</sup>*<sup>0</sup> <sup>∈</sup> <sup>N</sup>, *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> and *<sup>y</sup>* <sup>∈</sup> [0, <sup>∞</sup>[. The lower bound *E L*,*tan y*,*X*0,*n* of the Kullback-Leibler information divergence (relative entropy) is derived by using *φ U <sup>λ</sup>* ≡ *φ* tan *λ*,*y* (cf. (52)), which corresponds to the tangent line of *<sup>φ</sup><sup>λ</sup>* at *<sup>y</sup>*, as a linear upper bound for *<sup>φ</sup><sup>λ</sup>* (*<sup>λ</sup>* ∈]0, 1[). More precisely, one gets *φ U λ* (*x*) := (*p U <sup>λ</sup>* − *αλ*) + (*q U <sup>λ</sup>* − *<sup>β</sup>λ*) *<sup>x</sup>* (*<sup>x</sup>* ∈ [0, <sup>∞</sup>[) with *<sup>p</sup><sup>λ</sup>* := *<sup>p</sup>λ*(*y*) := *<sup>φ</sup>λ*(*y*) − *<sup>y</sup><sup>φ</sup>* 0 *λ* (*y*) + *α<sup>λ</sup>* and *q<sup>λ</sup>* := *qλ*(*y*) := *φ* 0 *λ* (*y*) + *βλ*, implying *q<sup>λ</sup>* > 0 because of Properties 3 (P17). Analogously to (A27) and (A28), we obtain from (38) and (40) the convergence lim*λ*%<sup>1</sup> *<sup>B</sup> U <sup>λ</sup>*,*X*0,*<sup>n</sup>* = <sup>1</sup> and thus

$$I(P\_{\mathcal{A},\mathbb{H}}||P\_{\mathcal{H},\mathbb{H}}) \geq \lim\_{\lambda \nearrow 1} \left[ \frac{\partial \, a\_{\mathbb{H}}^{(q\_{\lambda})}}{\partial \lambda} X\_{0} + \left( \frac{\partial}{\partial \lambda} \frac{p\_{\lambda}}{q\_{\lambda}} \right) \sum\_{k=1}^{\mathfrak{n}} a\_{k}^{(q\_{\lambda})} + \frac{p\_{\lambda}}{q\_{\lambda}} \sum\_{k=1}^{\mathfrak{n}} \frac{\partial a\_{k}^{(q\_{\lambda})}}{\partial \lambda} + n \frac{\partial}{\partial \lambda} \left( \frac{p\_{\lambda}}{q\_{\lambda}} \mathcal{G}\_{\lambda} - a\_{\lambda} \right) \right]. \tag{A29}$$

As before, we compute the involved derivatives. From (30) to (32) as well as (P17) we get

$$\begin{split} \frac{\partial p\_{\lambda}}{\partial \lambda} &= \left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right)^{\lambda} f\_{\mathcal{H}}(y) \log\left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right) - \beta\_{\mathcal{A}} y \left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right)^{\lambda - 1} - \lambda \beta\_{\mathcal{A}} y \left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right)^{\lambda - 1} \log\left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right) \\ &+ \beta\_{\mathcal{H}} y \left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right)^{\lambda} - (1 - \lambda) \beta\_{\mathcal{H}} y \left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right)^{\lambda} \log\left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right) \\ \stackrel{\lambda \nearrow 1}{\longrightarrow} a\_{\mathcal{A}} \log\left(\frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)}\right) + \frac{y \cdot (a\_{\mathcal{A}} \beta\_{\mathcal{H}} - a\_{\mathcal{H}} \beta\_{\mathcal{A}})}{f\_{\mathcal{H}}(y)}, \end{split} \tag{A30}$$

and

$$\begin{split} \frac{\partial q\_{\lambda}}{\partial \lambda} &= & \beta\_{\mathcal{A}} \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right)^{\lambda - 1} + \lambda \beta\_{\mathcal{A}} \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right)^{\lambda - 1} \log \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right) - \beta\_{\mathcal{H}} \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right)^{\lambda} \\ &+ (1 - \lambda) \beta\_{\mathcal{H}} \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right)^{\lambda} \log \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right) \\ \stackrel{\lambda \nearrow 1}{\longrightarrow} & \beta\_{\mathcal{A}} \left( 1 + \log \left( \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \right) \right) - \beta\_{\mathcal{H}} \frac{f\_{\mathcal{A}}(y)}{f\_{\mathcal{H}}(y)} \quad =: \quad !. \end{split} \tag{A31}$$

Combining these two limits we get

$$\begin{split} \frac{\partial}{\partial\lambda} \left( \frac{p\_{\lambda}}{q\_{\lambda}} \boldsymbol{\beta}\_{\lambda} - \boldsymbol{a}\_{\lambda} \right) &= \quad \frac{q\_{\lambda} \left( \frac{\partial p\_{\lambda}}{\partial\lambda} \right) - p\_{\lambda} \left( \frac{\partial q\_{\lambda}}{\partial\lambda} \right)}{(q\_{\lambda})^{2}} \cdot \boldsymbol{\beta}\_{\lambda} + \frac{p\_{\lambda}}{q\_{\lambda}} \cdot (\boldsymbol{\beta}\_{\mathcal{A}} - \boldsymbol{\beta}\_{\mathcal{H}}) - (\boldsymbol{a}\_{\mathcal{A}} - \boldsymbol{a}\_{\mathcal{H}}) \\ \stackrel{\lambda \nearrow 1}{\longrightarrow} & \quad \left[ \frac{\boldsymbol{y} \cdot (\boldsymbol{a}\_{\mathcal{A}} \boldsymbol{\beta} \boldsymbol{\gamma} - \boldsymbol{a}\_{\mathcal{H}} \boldsymbol{\beta}\_{\mathcal{A}})}{f\_{\mathcal{H}}(\boldsymbol{y})} - \boldsymbol{a}\_{\mathcal{A}} \left( 1 - \frac{\boldsymbol{\beta}\_{\mathcal{H}} f\_{\mathcal{A}}(\boldsymbol{y})}{\boldsymbol{\beta}\_{\mathcal{A}} f\_{\mathcal{H}}(\boldsymbol{y})} \right) \right] + \boldsymbol{a}\_{\mathcal{H}} - \frac{\boldsymbol{a}\_{\mathcal{A}} \boldsymbol{\beta}\_{\mathcal{H}}}{\boldsymbol{\beta}\_{\mathcal{A}}}. \end{split} \tag{A.32}$$

The above calculation also implies that lim*λ*%<sup>1</sup> *∂ ∂λ pλ qλ* is finite and thus lim*λ*%<sup>1</sup> *∂ ∂λ pλ qλ* ∑ *n k*=1 *a* (*qλ*) *<sup>k</sup>* = 0 by means of Lemma A2. The proof of *<sup>I</sup>*(*P*A,*n*||*P*H,*n*) <sup>≥</sup> *<sup>E</sup> L*,tan *y*,*X*0,*n* is finished by using Lemma A3 with *l* defined in (A31) and by plugging the limits (A30) to (A32) in (A29).

To derive the lower bound *E L*,sec *k*,*X*0,*n* (cf. (73)) for fixed *<sup>k</sup>* ∈ N0, we use as a linear upper bound *<sup>φ</sup> U λ* for *<sup>φ</sup>λ*(·) (*<sup>λ</sup>* ∈]0, 1[) the secant line *<sup>φ</sup>* sec *λ*,*k* (cf. (53)) of *φ<sup>λ</sup>* across its arguments *k* and *k* + 1, corresponding to the choices *p<sup>λ</sup>* := *p* sec *<sup>λ</sup>*,*<sup>k</sup>* = (*<sup>k</sup>* + <sup>1</sup>) · *<sup>φ</sup>λ*(*k*) − *<sup>k</sup>* · *<sup>φ</sup>λ*(*<sup>k</sup>* + <sup>1</sup>) + *<sup>α</sup><sup>λ</sup>* and *<sup>q</sup><sup>λ</sup>* := *<sup>q</sup>* sec *λ*,*k* := *<sup>φ</sup>λ*(*<sup>k</sup>* + 1) − *<sup>φ</sup>λ*(*k*) + *<sup>β</sup>λ*, implying *q<sup>λ</sup>* > 0 because of Properties 3 (P18). As a side remark, notice that this *φ U λ* (*x*) may become positive for some *<sup>x</sup>* ∈ [0, <sup>∞</sup>[ (which is not always consistent with Goal (G1) for fixed *<sup>λ</sup>*, but leads to a tractable limit bound as *<sup>λ</sup>* tends to 1). Analogously to (A27) and (A28) we get again lim*λ*%<sup>1</sup> *<sup>B</sup> U <sup>λ</sup>*,*X*0,*<sup>n</sup>* = <sup>1</sup>, which leads to the lower bound given in (A29) with appropriately plugged-in quantities. As in the above proof of the lower bound *E L*,*tan y*,*X*0,*n* , the inequality *<sup>I</sup>*(*P*A,*n*||*P*H,*n*) <sup>≥</sup> *<sup>E</sup> L*,sec *k*,*X*0,*n* follows straightforwardly from Lemma A2, Lemma A3 and the three limits

*∂p<sup>λ</sup> ∂λ* <sup>=</sup> *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) *λ <sup>f</sup>*H(*k*) · (*k*+1)log *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) − *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) *λ <sup>f</sup>*H(*k*+1) · *<sup>k</sup>* log *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) *λ*%1 −→ *<sup>f</sup>*A(*k*)(*k*+1)log *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) <sup>−</sup> *<sup>f</sup>*A(*k*+1)*<sup>k</sup>* log *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) , *∂q<sup>λ</sup> ∂λ* <sup>=</sup> *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) *λ <sup>f</sup>*H(*k*+1)log *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) − *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) *λ <sup>f</sup>*H(*k*)log *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) *λ*%1 −→ *<sup>f</sup>*A(*k*+1)log *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) <sup>−</sup> *<sup>f</sup>*A(*k*)log *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) =: *l* , and *∂ ∂λ pλ qλ β<sup>λ</sup>* − *α<sup>λ</sup>* = *qλ ∂p<sup>λ</sup> ∂λ* − *p<sup>λ</sup> ∂q<sup>λ</sup> ∂λ* (*qλ*) 2 · *β<sup>λ</sup>* + *pλ qλ* · (*β*<sup>A</sup> − *β*H) − (*α*<sup>A</sup> − *α*H) *λ*%1 −→ *<sup>f</sup>*A(*k*)log *<sup>f</sup>*A(*k*) *<sup>f</sup>*H(*k*) *k*+<sup>1</sup> <sup>+</sup> *α*A *β*A <sup>−</sup> *<sup>f</sup>*A(*k*+1)log *<sup>f</sup>*A(*k*+1) *<sup>f</sup>*H(*k*+1) *<sup>k</sup>* <sup>+</sup> *α*A *β*A − *α*A*β*<sup>H</sup> *β*A + *<sup>α</sup>*H.

To construct the third lower bound *E L*,*hor X*0,*n* (cf. (74)), we start by using the horizontal line *φ* hor *λ* (·) (cf. (54)) as an upper bound of *<sup>φ</sup>λ*. For each fixed *<sup>λ</sup>* <sup>∈</sup>]0, 1[, it is defined by the intercept sup*x*∈N<sup>0</sup> *<sup>φ</sup>λ*(*x*). On PSP,3a ∪ PSP,3b, this supremum is achieved at the finite integer point *<sup>z</sup>* ∗ *λ* := arg max*x*∈N<sup>0</sup> *<sup>φ</sup>λ*(*x*) (since lim*x*→<sup>∞</sup> *<sup>φ</sup>λ*(*x*) = −∞) and there holds *<sup>φ</sup>λ*(*<sup>z</sup>* ∗ *λ* ) < 0 which leads with the parameters *q<sup>λ</sup>* = *βλ*, *p<sup>λ</sup>* = *φλ*(*z* ∗ *λ* ) + *α<sup>λ</sup>* to the Hellinger integral upper bound *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* <sup>=</sup> exp *φλ*(*z* ∗ *λ* ) · *n* < 1 (cf. Remark 1 (b)). We strive for computing the limit lim*λ*%<sup>1</sup> 1−*B U λ*,*X*0 ,*n λ*(1−*λ*) , which is not straightforward to solve since in general it seems to be intractable to express *z* ∗ *λ* explicitly in terms of *λ*. To circumvent this problem, we notice that it is sufficient to determine *z* ∗ *λ* in a small *<sup>e</sup>*−environment ]<sup>1</sup> <sup>−</sup> *<sup>e</sup>*, 1[. To accomplish this, we incorporate lim*λ*%<sup>1</sup> *<sup>φ</sup>λ*(*x*) = <sup>0</sup> for all *<sup>x</sup>* ∈ [0, <sup>∞</sup>[ and calculate by using l'Hospital's rule

$$\lim\_{\lambda \nearrow 1} \frac{\phi\_{\lambda}(\mathbf{x})}{1 - \lambda} = (\alpha\_{\mathcal{A}} + \beta\_{\mathcal{A}}\mathbf{x}) \left[ -\log \left( \frac{\alpha\_{\mathcal{A}} + \beta\_{\mathcal{A}}\mathbf{x}}{\alpha\_{\mathcal{H}} + \beta\_{\mathcal{H}}\mathbf{x}} \right) + 1 \right] - (\alpha\_{\mathcal{H}} + \beta\_{\mathcal{H}}\mathbf{x}).$$

Accordingly, let us define *z* ∗ := arg max*x*∈N<sup>0</sup> n (*α*<sup>A</sup> + *<sup>β</sup>*A*x*) h <sup>−</sup> log *α*A+*β*A*x α*H+*β*<sup>H</sup> *x* + 1 i − (*α*<sup>H</sup> + *<sup>β</sup>*H*x*) o (note that the maximum exists since lim*x*→<sup>∞</sup> n (*α*<sup>A</sup> + *<sup>β</sup>*A*x*) h <sup>−</sup> log *α*A+*β*A*x α*H+*β*<sup>H</sup> *x* + 1 i − (*α*<sup>H</sup> + *<sup>β</sup>*H*x*) o = −∞). Due to continuity of the function (*λ*, *<sup>x</sup>*) 7→ *<sup>φ</sup>λ*(*x*) 1−*λ* , there exists an *<sup>e</sup>* > 0 such that for all *<sup>λ</sup>* ∈]1 − *<sup>e</sup>*, 1[ there holds *z* ∗ *<sup>λ</sup>* = *z* ∗ . Applying these considerations, we get with l'Hospital's rule

$$I(P\_{\mathcal{A},\mathbb{H}}||P\_{\mathcal{H},\mathbb{H}}) \ge \lim\_{\lambda \nearrow 1} \frac{1 - \exp\left\{\phi\_{\lambda}(\mathbf{z}^\*) \cdot n\right\}}{\lambda(1 - \lambda)} \\ = \left[f\_{\mathcal{A}}(\mathbf{z}^\*) \cdot \left[\log\left(\frac{f\_{\mathcal{A}}(\mathbf{z}^\*)}{f\_{\mathcal{H}}(\mathbf{z}^\*)}\right) - 1\right] + f\_{\mathcal{H}}(\mathbf{z}^\*)\right] \cdot n \\ \ge 0. \tag{A33}$$

In fact, for the current parameter constellation PSP,3a ∪ PSP,3b we have *<sup>φ</sup>λ*(*x*) < 0 for all *<sup>λ</sup>* ∈]0, 1[ and all *<sup>x</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> which implies *<sup>f</sup>*A(*<sup>z</sup>* ∗ ) 6= *<sup>f</sup>*H(*<sup>z</sup>* ∗ ) by Lemma A1; thus, we even get *E L*,*hor <sup>X</sup>*0,*<sup>n</sup>* > <sup>0</sup> for all *<sup>n</sup>* ∈ N by virtue of the inequality <sup>−</sup> log *f*H(*z* ∗ ) *f*A(*z* ∗) > − *f*H(*z* ∗ ) *f*A(*z* ∗) + 1.

For the case PSP,2, the above-mentioned procedure leads to *<sup>z</sup>* ∗ *<sup>λ</sup>* = 0 = *z* ∗ (*<sup>λ</sup>* ∈]0, 1[) which implies *φλ*(*z* ∗ *λ* ) = 0, *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ <sup>1</sup> and thus the trivial lower bound *<sup>E</sup> L*,*hor <sup>X</sup>*0,*<sup>n</sup>* = lim*λ*%<sup>1</sup> 1−*B U λ*,*X*0 ,*n λ*(1−*λ*) = <sup>0</sup> follows for all *<sup>n</sup>* ∈ N. In contrast, for the case PSP,3c one gets *<sup>z</sup>* ∗ *<sup>λ</sup>* = *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> = *z* ∗ (*<sup>λ</sup>* ∈]0, 1[) which nevertheless also implies *φλ*(*z* ∗ *λ* ) = 0 and hence *E L*,*hor <sup>X</sup>*0,*<sup>n</sup>* <sup>≡</sup> <sup>0</sup>. On <sup>P</sup>SP,4, we have sup*x*∈N<sup>0</sup> *<sup>φ</sup>λ*(*x*) = lim*x*→<sup>∞</sup> *<sup>φ</sup>λ*(*x*) = <sup>0</sup> and hence we set *E L*,*hor <sup>X</sup>*0,*<sup>n</sup>* ≡ <sup>0</sup>.

To show the strict positivity *E L <sup>X</sup>*0,*<sup>n</sup>* > 0 in the parameter case PSP,2, we inspect the bound *<sup>E</sup> L*,*sec* 0,*X*0,*n* . With *<sup>α</sup>* :<sup>=</sup> *<sup>α</sup>*• :<sup>=</sup> *<sup>α</sup>*<sup>A</sup> <sup>=</sup> *<sup>α</sup>*<sup>H</sup> (the bullet will be omitted in this proof) and the auxiliary variable *<sup>x</sup>* :<sup>=</sup> *β*H *β*A > 0, the definition (73) respectively its special case (76) rewrites for all *<sup>n</sup>* ∈ N as

*E L*,*sec* 0,*X*0,*n* := *E L*,*sec* 0,*X*0,*n* (*x*) := h <sup>−</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>β</sup>*A) · log *α*+*β*A*x α*+*β*<sup>A</sup> + *<sup>β</sup>*A(*<sup>x</sup>* − <sup>1</sup>) i · 1−(*β*A) *n* 1−*β*<sup>A</sup> · h *<sup>X</sup>*<sup>0</sup> <sup>−</sup> *<sup>α</sup>* 1−*β*<sup>A</sup> i + h *α β*A(1−*β*A) <sup>−</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>β</sup>*A) · log *α*+*β*A*x α*+*β*<sup>A</sup> + *<sup>β</sup>*A(*<sup>x</sup>* − <sup>1</sup>) + *<sup>α</sup> β*A (*<sup>α</sup>* <sup>+</sup> *<sup>β</sup>*A) · log *α*+*β*A*x α*+*β*<sup>A</sup> − *α*(*x* − 1) i · *<sup>n</sup>* , if *<sup>β</sup>*<sup>A</sup> 6= 1, h <sup>−</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) · log *α*+*x α*+1 + *x* − 1 i · - *α* 2 · *n* <sup>2</sup> + *X*<sup>0</sup> + *<sup>α</sup>* 2 · *n* + h (*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) · log *α*+*x α*+1 − *x* + 1 i · *<sup>α</sup>* · *<sup>n</sup>* , if *<sup>β</sup>*<sup>A</sup> = 1. (A34)

To prove that *E L*,*sec* 0,*X*0,*<sup>n</sup>* > <sup>0</sup> for all *<sup>X</sup>*<sup>0</sup> ∈ N and all *<sup>n</sup>* ∈ N it suffices to show that *<sup>E</sup> L*,*sec* 0,*X*0,*n* (1) = *∂ ∂x E L*,*sec* 0,*X*0,*n* (1) = 0 and *∂* 2 *∂x* <sup>2</sup> *E L*,*sec* 0,*X*0,*n* (*x*) > <sup>0</sup> for all *<sup>x</sup>* ∈]0, <sup>∞</sup>[\{1}. The assertion *<sup>E</sup> L*,*sec* 0,*X*0,*n* (1) = 0 is trivial from (A34). Moreover, we obtain *n*

$$\left(\frac{\partial}{\partial \mathbf{x}} E\_{0, \mathbf{X} \boldsymbol{\alpha}, \boldsymbol{n}}^{L, \mathbf{x} \boldsymbol{\alpha}}\right)(\mathbf{x}) = \begin{cases} \beta\_{\mathcal{A}} \cdot \left[1 - \frac{\boldsymbol{a} + \beta\_{\mathcal{A}}}{\boldsymbol{a} + \beta\_{\mathcal{A}} \boldsymbol{\alpha}}\right] \cdot \frac{1 - (\beta\_{\mathcal{A}})^{\boldsymbol{n}}}{1 - \beta\_{\mathcal{A}}} \cdot \left[\mathbf{X}\_{0} - \frac{\boldsymbol{a}}{1 - \beta\_{\mathcal{A}}}\right] \\\\ + \boldsymbol{a} \cdot \left(1 - \frac{\boldsymbol{a} + \beta\_{\mathcal{A}}}{\boldsymbol{a} + \beta\_{\mathcal{A}} \boldsymbol{\alpha}}\right) \cdot \frac{\beta\_{\mathcal{A}}}{1 - \beta\_{\mathcal{A}}} \cdot \boldsymbol{n} \, \boldsymbol{\prime} & \text{if } \beta\_{\mathcal{A}} \neq 1 \boldsymbol{\alpha} \\\\ \left[1 - \frac{\boldsymbol{a} + \mathbf{1}}{\boldsymbol{a} + \mathbf{x}}\right] \cdot \left[\frac{\boldsymbol{a}}{2} \cdot \boldsymbol{n}^{2} + \left(\mathbf{X}\_{0} - \frac{\boldsymbol{a}}{2}\right) \cdot \boldsymbol{n}\right] \, \boldsymbol{\alpha} & \text{if } \beta\_{\mathcal{A}} = 1 \boldsymbol{\alpha} \end{cases}$$

which immediately yields *∂ ∂x E L*,*sec* 0,*X*0,*n* (1) = 0. For the second derivative we get

$$\begin{pmatrix} \frac{\partial^2}{\partial x^2} \mathbf{E}\_{0, \mathbf{X}\_0, \mu}^{L, \text{sec}} \\\\ \frac{\partial}{\partial x^2} \mathbf{E}\_{0, \mathbf{X}\_0, \mu}^{L, \text{sec}} \end{pmatrix} (\mathbf{x}) = \begin{cases} \begin{cases} \frac{(a + \beta\_A)}{(a + \beta\_A x)^2} \cdot \frac{1 - (\beta\_A)^n}{1 - \beta\_A} \cdot \left[ \mathbf{X}\_0 - \frac{\mu}{1 - \beta\_A} \right] \\\\ + a \frac{a + \beta\_A}{(a + \beta\_A x)^2} \cdot \frac{\beta\_A^2}{1 - \beta\_A} \cdot n > 0, & \text{if } \beta\_A \neq 1, \\\\ \frac{a + 1}{(a + \mu)^2} \cdot \left[ \frac{\mu}{2} \cdot n^2 + \left( \mathbf{X}\_0 - \frac{\mu}{2} \right) \cdot n \right] > 0, & \text{if } \beta\_A = 1, \end{cases} \end{cases} \tag{A35}$$

where the strict positivity of *E L*,*sec* 0,*X*0,*n* in the case *<sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> <sup>1</sup> follows immediately by replacing *<sup>X</sup>*<sup>0</sup> with <sup>0</sup> and by using the obvious relation <sup>1</sup> 1−*β*<sup>A</sup> · h *n* − 1−*β n* A 1−*β*<sup>A</sup> i = <sup>1</sup> 1−*β*<sup>A</sup> ∑ *n*−1 *k*=0 1 − *β k* A > 0. The strict positivity in the case *<sup>β</sup>*<sup>A</sup> <sup>=</sup> <sup>1</sup> is trivial by inspection.

For the constellation <sup>P</sup>SP,4 with parameters *<sup>β</sup>* :<sup>=</sup> *<sup>β</sup>*• :<sup>=</sup> *<sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*H, *<sup>α</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>α</sup>*H, the strict positivity of *E L <sup>X</sup>*0,*<sup>n</sup>* > 0 follows by showing that *<sup>E</sup> L*,*tan y*,*X*0,*n* converges from above to zero as *y* tends to infinity. This is done by proving lim*y*→<sup>∞</sup> *<sup>y</sup>* · *<sup>E</sup> L*,*tan y*,*X*0,*n* ∈]0, <sup>∞</sup>[. To see this, let us first observe that by l'Hospital's rule we get

$$\lim\_{y \to \infty} y \cdot \log \left( \frac{\mathfrak{a}\_{\mathcal{A}} + \beta y}{\mathfrak{a}\_{\mathcal{H}} + \beta y} \right) = \frac{\mathfrak{a}\_{\mathcal{A}} - \mathfrak{a}\_{\mathcal{H}}}{\beta} \qquad \text{as well as} \qquad \lim\_{y \to \infty} y \cdot \left( 1 - \frac{\mathfrak{a}\_{\mathcal{A}} + \beta y}{\mathfrak{a}\_{\mathcal{H}} + \beta y} \right) = -\frac{\mathfrak{a}\_{\mathcal{A}} - \mathfrak{a}\_{\mathcal{H}}}{\beta}.$$

From this and (72), we obtain lim*y*→<sup>∞</sup> *<sup>y</sup>* · *<sup>E</sup> L*,*tan <sup>y</sup>*,*X*0,*<sup>n</sup>* = (*α*A−*α*H) 2 *β* · *<sup>n</sup>* > <sup>0</sup> in both cases *<sup>β</sup>* 6= <sup>1</sup> and *<sup>β</sup>* = <sup>1</sup>.

Finally, for the parameter case PSP,3c we consider the bound *<sup>E</sup> L*,*tan y* ∗ ,*X*0,*n* , with *y* ∗ = *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> . Since *<sup>α</sup>*<sup>A</sup> + *<sup>β</sup>*A*<sup>y</sup>* <sup>∗</sup> = *<sup>α</sup>*<sup>H</sup> + *<sup>β</sup>*H*<sup>y</sup>* ∗ , it is easy to see that *E L*,*tan y* ∗ ,*X*0,*<sup>n</sup>* = <sup>0</sup> for all *<sup>n</sup>* ∈ N. However, the condition *∂ ∂y E L*,*tan y*,*X*0,*n* (*y* ∗ ) <sup>6</sup><sup>=</sup> <sup>0</sup> implies that sup*y*≥<sup>0</sup> *E L*,*tan <sup>y</sup>*,*X*0,*<sup>n</sup>* > <sup>0</sup>. The explicit form (75) of this condition follows from

$$\begin{pmatrix} \frac{\partial}{\partial y} E\_{y,\mathcal{X}\_0,\mu}^{L,\text{fan}} \end{pmatrix} \begin{pmatrix} y \\ \end{pmatrix} = \begin{cases} \frac{\left(\frac{\alpha\_A \beta\_\mathcal{H} - \alpha\_\mathcal{H} \beta\_\mathcal{A}}{f\_A(y) \left(f\_\mathcal{H} \left(y\right)\right)^2} \cdot \frac{1 - \left(\beta\_\mathcal{A}\right)^n}{1 - \beta\_\mathcal{A}} \cdot \left[X\_0 - \frac{\alpha\_\mathcal{A}}{1 - \beta\_\mathcal{A}}\right]}{1 - \beta\_\mathcal{A}} \\ \quad + \frac{\frac{\alpha\_A \beta\_\mathcal{H} - \alpha\_\mathcal{H} \beta\_\mathcal{A}}{\left(f\_\mathcal{H} \left(y\right)\right)^2} \cdot \left[\frac{\alpha\_A}{\beta\_\mathcal{A} \left(1 - \beta\_\mathcal{A}\right) f\_\mathcal{A} \left(y\right)} - \frac{\alpha\_\mathcal{A} \beta\_\mathcal{H} - \alpha\_\mathcal{H} \beta\_\mathcal{A}}{\beta\_\mathcal{A}}\right]}{\beta\_\mathcal{A}} \right) \cdot n \,\,, \qquad \text{if } \beta\_\mathcal{A} \neq 1, \\\quad \frac{\left(\frac{\alpha\_A \beta\_\mathcal{H} - \alpha\_\mathcal{H}}{f\_\mathcal{A} \left(y\right) \left(f\_\mathcal{H} \left(y\right)\right)^2} \cdot \left[\frac{\alpha\_\mathcal{A}}{2} \cdot n^2 + \left(X\_0 + \frac{\alpha\_\mathcal{A}}{2}\right) \cdot n\right] - \frac{\left(\frac{\alpha\_A \beta\_\mathcal{H} - \alpha\_\mathcal{H}}{\beta\_\mathcal{A}}\right)^2}{\left(f\_\mathcal{H} \left(y\right)\right)^2} \cdot n \,\,, \quad \text{if } \beta\_\mathcal{A} = 1, \end{cases}$$

*<sup>y</sup>* ≥ <sup>0</sup>, by using the particular choice *<sup>y</sup>* = *<sup>y</sup>* ∗ together with *<sup>f</sup>*A(*<sup>y</sup>* ∗ ) = *<sup>f</sup>*H(*<sup>y</sup>* ∗ ) = − *α*A*β*H−*α*<sup>H</sup> *β*<sup>A</sup> *β*A−*β*<sup>H</sup> .

#### *Appendix A.3. Proofs and Auxiliary Lemmas for Section 6*

**Proof of Lemma 2.** A closed-form representation of a sequence (e*an*)*n*∈N<sup>0</sup> defined in (83) to (85) is given by the formula

$$\widetilde{a}\_n = \sum\_{k=0}^{n-1} \left( \mathfrak{c} + \rho\_k \right) d^{n-1-k}. \tag{A36}$$

This can be seen by induction: from (83) we obtain with <sup>e</sup>*a*<sup>0</sup> <sup>=</sup> <sup>0</sup> for the first element <sup>e</sup>*a*<sup>1</sup> <sup>=</sup> *<sup>c</sup>* <sup>+</sup> *<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> ∑ 0 *k*=0 (*c* + *ρ<sup>k</sup>* )*d* −*k* . Supposing that (A36) holds for the *n*-th element, the induction step is

$$\widetilde{a}\_{n+1} = \left. \mathcal{c} + d \cdot \widetilde{a}\_{\mathfrak{n}} + \rho\_{\mathfrak{n}} \right| \\ = \left. \mathcal{c} + d \cdot \sum\_{k=0}^{n-1} \left( \mathcal{c} + \rho\_{\mathbb{k}} \right) d^{n-1-k} + \rho\_{\mathfrak{n}} \\ = \sum\_{k=0}^{n} \left( \mathcal{c} + \rho\_{\mathbb{k}} \right) d^{n-k} \dots$$

In order to obtain the explicit representation of <sup>e</sup>*an*, we consider first the case <sup>0</sup> <sup>≤</sup> *<sup>ν</sup>* <sup>&</sup>lt; <sup>κ</sup> <sup>&</sup>lt; *<sup>d</sup>* and *ρ<sup>n</sup>* = *K*<sup>1</sup> · κ *<sup>n</sup>* <sup>+</sup> *<sup>K</sup>*<sup>2</sup> · *<sup>ν</sup> n* , which leads to

$$\begin{split} \widetilde{a}\_{n} &= \quad d^{n-1} \sum\_{k=0}^{n-1} \left( c \cdot d^{-k} + K\_{1} \cdot \left( \frac{\varkappa}{d} \right)^{k} + K\_{2} \cdot \left( \frac{\nu}{d} \right)^{k} \right) \\ &= \quad d^{n-1} \cdot \left[ c \cdot \frac{1 - d^{-n}}{1 - d^{-1}} + K\_{1} \cdot \frac{1 - \left( \frac{\varkappa}{d} \right)^{n}}{1 - \frac{\varkappa}{d}} + K\_{2} \cdot \frac{1 - \left( \frac{\nu}{d} \right)^{n}}{1 - \frac{\nu}{d}} \right] \\ &= \quad \frac{c}{1 - d} (1 - d^{n}) + K\_{1} \cdot \frac{d^{n} - \varkappa^{n}}{d - \varkappa} + K\_{2} \cdot \frac{d^{n} - \nu^{n}}{d - \nu}. \end{split} \tag{A37}$$

Hence, for the corresponding sum we get

$$\begin{split} \sum\_{k=1}^{n} \tilde{a}\_{k} &= \quad \sum\_{k=1}^{n} \left[ \frac{c}{1-d} + \left( \frac{K\_{1}}{d-\varkappa} + \frac{K\_{2}}{d-\nu} - \frac{c}{1-d} \right) \cdot d^{k} - \frac{K\_{1}}{d-\varkappa} \cdot \varkappa^{k} - \frac{K\_{2}}{d-\nu} \cdot \nu^{k} \right] \\ &= \quad \frac{c}{1-d} \cdot n + \left( \frac{K\_{1}}{d-\varkappa} + \frac{K\_{2}}{d-\nu} - \frac{c}{1-d} \right) \cdot \frac{d \cdot (1-d^{n})}{1-d} - \frac{K\_{1} \cdot \varkappa \cdot (1-\varkappa^{n})}{(d-\varkappa)(1-\varkappa)} - \frac{K\_{2} \cdot \nu \cdot (1-\nu^{n})}{(d-\nu)(1-\nu)}. \end{split} \tag{A38}$$

Consider now the case 0 ≤ *<sup>ν</sup>* < κ = *<sup>d</sup>*. Then some expressions in (A37) and (A38) have a zero denominator. In this case, the evaluation of (A36) becomes

$$\begin{split} \widetilde{a}\_{n} &= & d^{n-1} \sum\_{k=0}^{n-1} \left( c \cdot d^{-k} + K\_{1} + K\_{2} \cdot \left( \frac{\nu}{d} \right)^{k} \right) = d^{n-1} \cdot \left[ c \cdot \frac{1 - d^{-n}}{1 - d^{-1}} + K\_{1} \cdot n + K\_{2} \cdot \frac{1 - \left( \frac{\nu}{d} \right)^{n}}{1 - \frac{\nu}{d}} \right] \\ &= & \frac{c}{1 - d} (1 - d^{n}) + K\_{1} \cdot n \cdot d^{n-1} + K\_{2} \cdot \frac{d^{n} - \nu^{n}}{d - \nu}. \end{split} \tag{A.39}$$

Before we calculate the corresponding sum ∑ *n k*=1 e*ak* , we notice that

$$\sum\_{k=1}^{n} k \cdot d^{k-1} = \sum\_{k=1}^{n} \frac{\partial}{\partial d} d^k = \frac{\partial}{\partial d} \sum\_{k=1}^{n} d^k = \frac{\partial}{\partial d} \left( \frac{d \cdot (1 - d^n)}{1 - d} \right) = \frac{1 - n \cdot d^n (1 - d) - d^n}{(1 - d)^2}.$$

Using this fact, we obtain

$$\begin{split} \sum\_{k=1}^{n} \tilde{a}\_{k} &= \sum\_{k=1}^{n} \left[ \frac{c}{1-d} (1-d^{k}) + K\_{1} \cdot k \cdot d^{k-1} + K\_{2} \cdot \frac{d^{k} - v^{k}}{d - v} \right] \\ &= \frac{c}{1-d} \cdot n + \sum\_{k=1}^{n} \left( \frac{K\_{2}}{d - \nu} - \frac{c}{1-d} \right) d^{k} + K\_{1} \sum\_{k=1}^{n} k \cdot d^{k-1} - \frac{K\_{2}}{d - \nu} \sum\_{k=1}^{n} \nu^{k} \\ &= \left( \frac{K\_{2}}{d - \nu} - \frac{c}{1-d} \right) \frac{d \cdot (1 - d^{n})}{1 - d} + K\_{1} \cdot \frac{1 - n \cdot d^{n} (1 - d) - d^{n}}{(1 - d)^{2}} - \frac{K\_{2} \cdot \nu (1 - \nu^{n})}{(d - \nu)(1 - \nu)} + \frac{c}{1 - d} \cdot n \\ &= \left( \frac{K\_{1}}{d(1 - d)} + \frac{K\_{2}}{d - \nu} - \frac{c}{1 - d} \right) \frac{d \cdot (1 - d^{n})}{1 - d} - \frac{K\_{2} \cdot \nu (1 - \nu^{n})}{(d - \nu)(1 - \nu)} + \left( \frac{c}{1 - d} - \frac{K\_{1} \cdot d^{n}}{1 - d} \right) \cdot n. \end{split}$$

**Proof of Lemma 3.** (a) In this case we have 0 < *q* < *βλ*. To prove part (i), we consider the function *ξ* (*q*) *λ* (·) on [*x* (*q*) 0 , 0], the range of the sequence *a* (*q*) *n n*∈N (recall Properties 1 (P1)). For tackling the left-hand inequality in (i), we compare *ξ* (*q*) *λ* (*x*) = *q* · *e <sup>x</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* with the quadratic function

$$\underline{\mathbf{Y}}\_{\lambda}^{(q)}(\mathbf{x}) := \frac{q}{2} \boldsymbol{\varepsilon}^{x\_0^{(q)}} \cdot \mathbf{x}^2 + q \boldsymbol{\varepsilon}^{x\_0^{(q)}} \left(\mathbf{1} - \mathbf{x}\_0^{(q)}\right) \cdot \mathbf{x} + \mathbf{x}\_0^{(q)} \left(\mathbf{1} - q \boldsymbol{\varepsilon}^{x\_0^{(q)}} + \frac{q}{2} \boldsymbol{\varepsilon}^{x\_0^{(q)}} \mathbf{x}\_0^{(q)}\right). \tag{A40}$$

Clearly, one has the relations Υ (*q*) *λ* (*x* (*q*) 0 ) = *x* (*q*) <sup>0</sup> = *ξ* (*q*) *λ* (*x* (*q*) 0 ), Υ (*q*)0 *λ* (*x* (*q*) 0 ) = *q* · *e x* (*q*) <sup>0</sup> = *ξ* (*q*)0 *λ* (*x* (*q*) 0 ), and Υ (*q*)00 *λ* (*x*) < *ξ* (*q*)00 *λ* (*x*) for all *<sup>x</sup>* ∈]*<sup>x</sup>* (*q*) 0 , 0]. Hence, Υ (*q*) *λ* (·) is on ]*<sup>x</sup>* (*q*) 0 , 0] a strict lower functional bound of *ξ* (*q*) *λ* (·). We are now ready to prove the left-hand inequality in (i) by induction. For *<sup>n</sup>* = 1, we easily see that *a* (*q*) <sup>1</sup> < *a* (*q*) 1 iff *x* (*q*) 0 <sup>1</sup> <sup>−</sup> *qe<sup>x</sup>* (*q*) <sup>0</sup> + *q* 2 *e x* (*q*) <sup>0</sup> *x* (*q*) 0 < *q* − *β<sup>λ</sup>* iff Υ (*q*) *λ* (0) < *ξ* (*q*) *λ* (0), and the latter is obviously true. Let us assume that *a* (*q*) *<sup>n</sup>* ≤ *a* (*q*) *<sup>n</sup>* holds. From this, (93), (78) and (80) we obtain

$$\begin{split} &0 < \underline{\rho}\_{n}^{(q)} = \frac{q}{2} \varepsilon^{\chi\_{0}^{(q)}} \left( \mathbf{x}\_{0}^{(q)} \cdot \left( q \cdot \mathbf{c}^{\chi\_{0}^{(q)}} \right)^{n} \right)^{2} = \frac{q}{2} \varepsilon^{\chi\_{0}^{(q)}} \left( a\_{n}^{(q),T} - \mathbf{x}\_{0}^{(q)} \right)^{2} \\ &< \frac{q}{2} \varepsilon^{\chi\_{0}^{(q)}} \left( a\_{n}^{(q)} - \mathbf{x}\_{0}^{(q)} \right)^{2} = \underline{\mathbf{Y}}\_{\lambda}^{(q)} \left( a\_{n}^{(q)} \right) - d^{(q),T} \cdot a\_{n}^{(q)} - \mathbf{x}\_{0}^{(q)} \cdot \left( 1 - d^{(q),T} \right) \\ &< \xi\_{\lambda}^{(q)} \left( a\_{n}^{(q)} \right) - d^{(q),T} \cdot a\_{n}^{(q)} - \mathbf{x}\_{0}^{(q)} \cdot \left( 1 - d^{(q),T} \right) \\ &< \left( a\_{n+1}^{(q),T} - d^{(q),T} \cdot \underline{a}\_{n}^{(q)} - \mathbf{x}\_{0}^{(q)} \right) \cdot \left( 1 - d^{(q),T} \right) = a\_{n+1}^{(q)} - \mathbf{f}\_{\lambda}^{(q),T} (\underline{a}\_{n}^{(q)}) \,. \end{split}$$

Thus, there holds *a* (*q*) *<sup>n</sup>*+<sup>1</sup> < *a* (*q*) *n*+1 . For the right-hand inequality in (i), we proceed analogously:

$$\nabla\_{\lambda}^{(q)}(\mathbf{x}) := \frac{q}{2} e^{\mathbf{x}\_0^{(q)}} \cdot \mathbf{x}^2 + \left(1 - \frac{q}{2} e^{\mathbf{x}\_0^{(q)}} \mathbf{x}\_0^{(q)} - \frac{q - \mathcal{G}\_{\lambda}}{\mathbf{x}\_0^{(q)}}\right) \cdot \mathbf{x} + q - \mathcal{g}\_{\lambda} \tag{A41}$$

satisfies Υ (*q*) *λ* (*x* (*q*) 0 ) = *x* (*q*) <sup>0</sup> = *ξ* (*q*) *λ* (*x* (*q*) 0 ), Υ (*q*) *λ* (0) = *q* − *β<sup>λ</sup>* = *ξ* (*q*) *λ* (0) as well as Υ (*q*)00 *λ* (*x*) < *ξ* (*q*)00 *λ* (*x*) for all *x* ∈]*x* (*q*) 0 , 0]. Hence, Υ (*q*) *λ* (·) is on ]*<sup>x</sup>* (*q*) 0 , 0] a strict upper functional bound of *ξ* (*q*) *λ* (·). Let us first observe the obvious relation *a* (*q*) <sup>1</sup> = *q* − *β<sup>λ</sup>* = *a* (*q*) <sup>1</sup> < 0, and assume that *a* (*q*) *<sup>n</sup>* ≥ *a* (*q*) *<sup>n</sup>* (*<sup>n</sup>* ∈ N) holds. From this, (95), (79), and (80) we obtain the desired inequality *a* (*q*) *<sup>n</sup>*+<sup>1</sup> > *a* (*q*) *n*+1 by

$$\begin{split} 0 &> \overline{\eta}\_{n}^{(q)} = -\Gamma\_{\leq}^{(q)} \left( d^{(q),T} \right)^{n} \cdot \frac{a\_{n}^{(q),S}}{x\_{0}^{(q)}} = \frac{q}{2} \, e^{x\_{0}^{(q)}} \left( a\_{n}^{(q),T} - x\_{0}^{(q)} \right) \cdot a\_{n}^{(q),S} \\ &\geq \frac{q}{2} \, e^{x\_{0}^{(q)}} \left( a\_{n}^{(q)} - x\_{0}^{(q)} \right) \cdot a\_{n}^{(q)} = \overline{\Upsilon}\_{\lambda}^{(q)} \left( a\_{n}^{(q)} \right) - d^{(q),S} \cdot a\_{n}^{(q)} - (q - \beta\_{\lambda}) \\ &> \xi\_{\lambda}^{(q)} \left( a\_{n}^{(q)} \right) - d^{(q),S} \cdot a\_{n}^{(q)} - (q - \beta\_{\lambda}) \geq a\_{n+1}^{(q)} - d^{(q),S} \cdot \overline{a}\_{n}^{(q)} - (q - \beta\_{\lambda}) = a\_{n+1}^{(q)} - \xi\_{\lambda}^{(q),S} (\overline{a}\_{n}^{(q)}) \ . \end{split}$$

The explicit representations of the sequences *a* (*q*) *n n*∈N , *a* (*q*) *n n*∈N and *a* (*q*) *n n*∈N follow from (86) by incorporating the appropriate constants mentioned in the prelude of Lemma 3. With (83) to (85) and (86) we immediately achieve *a* (*q*) *<sup>n</sup>* > *a* (*q*),*T <sup>n</sup>* for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Analogously, for all *<sup>n</sup>* <sup>≥</sup> <sup>2</sup>, we get *<sup>ρ</sup>n*−<sup>1</sup> <sup>&</sup>lt; <sup>0</sup>, which implies that *a* (*q*) *<sup>n</sup>* < *a* (*q*),*S <sup>n</sup>* for all *<sup>n</sup>* ≥ <sup>2</sup>. For *<sup>n</sup>* = <sup>1</sup> one obtains *<sup>ρ</sup>*<sup>0</sup> = <sup>0</sup> as well as *<sup>a</sup>* (*q*) <sup>1</sup> = *a* (*q*),*S* <sup>1</sup> = *a* (*q*) <sup>1</sup> = *<sup>q</sup>* − *<sup>β</sup>λ*.

For the second part (ii), we employ the representation (A36) which leads to

$$\underline{a}\_{n}^{(q)} = \sum\_{k=0}^{n-1} \left( d^{(q),T} \right)^{n-1-k} \cdot \left( \underline{\rho}\_{k}^{(q)} + \mathbf{x}\_{0}^{(q)} \cdot (1 - d^{(q),T}) \right),$$
as well as 
$$\overline{a}\_{n}^{(q)} = \sum\_{k=0}^{n-1} \left( d^{(q),S} \right)^{n-1-k} \cdot \left( \overline{\rho}\_{k}^{(q)} + (q - \beta\_{\lambda}) \right).$$

The strict decreasingness of both sequences follows from

$$\underline{\rho}\_{\lambda}^{(q)} + \mathbf{x}\_0^{(q)} (1 - d^{(q), T}) = \frac{q \underline{\rho}\_0^{(q)}}{2} \left( \mathbf{x}\_0^{(q)} \right)^2 \left( d^{(q), T} \right)^{2t} + \mathbf{x}\_0^{(q)} \left( 1 - d^{(q), T} \right) \\ \leq \underline{\mathbf{Y}}\_{\lambda}^{(q)} (0) \\ < \underline{\mathbf{y}}\_{\lambda}^{(q)} (0) \\ = q - \underline{\rho}\_{\lambda} \\ < 0$$

and from the fact that *ρ* (*q*) *<sup>k</sup>* ≤ 0 for all *<sup>k</sup>* ∈ N<sup>0</sup> and *<sup>q</sup>* < *<sup>β</sup>λ*. Part (iii) follows directly from (i), since *d* (*q*),*T* , *d* (*q*),*<sup>S</sup>* <sup>∈</sup>]0, 1[.

Let us now prove part (b), where max{0, *<sup>β</sup>λ*} <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; min <sup>n</sup> 1 , *e βλ*−1 o is assumed. To tackle part (i), we compare *ξ* (*q*) *λ* (*x*) = *q* · *e <sup>x</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* with the quadratic function

$$\underline{\mathbf{v}}\_{\lambda}^{(q)}(\mathbf{x}) := \frac{q}{2} \cdot \mathbf{x}^2 + q \cdot \left(\mathbf{e}^{\mathbf{x}\_0^{(q)}} - \mathbf{x}\_0^{(q)}\right) \cdot \mathbf{x} + \mathbf{x}\_0^{(q)} \left(1 - q \mathbf{e}^{\mathbf{x}\_0^{(q)}} + \frac{q}{2} \mathbf{x}\_0^{(q)}\right) \\ > 0 \tag{A42}$$

on the interval [0, *x* (*q*) 0 ]. Clearly, we have *υ* (*q*) *λ x* (*q*) 0 = *ξ* (*q*) *λ* (*x* (*q*) 0 ) = *x* (*q*) 0 , *υ* (*q*)0 *λ* (*x* (*q*) 0 ) = *ξ* (*q*)0 *λ* (*x* (*q*) 0 ) = *qe<sup>x</sup>* (*q*) 0 and 0 < *υ* (*q*)00 *λ* (*x*) < *ξ* (*q*)00 *λ* (*x*) for all *<sup>x</sup>* ∈]0, *<sup>x</sup>* (*q*) 0 ]. Thus, *υ* (*q*) *λ* (·) constitutes a positive functional lower bound for *ξ* (*q*) *λ* (·) on [0, *<sup>x</sup>* (*q*) 0 ]. Let us now prove the left-hand inequality of (i) by induction: for *n* = 1 we get *a* (*q*) <sup>1</sup> = *υ* (*q*) *λ* (0) < *ξ* (*q*) *λ* (0) = *a* (*q*) 1 . Moreover, by assuming *a* (*q*) *<sup>n</sup>* ≤ *a* (*q*) *<sup>n</sup>* for *<sup>n</sup>* ∈ N, we obtain with the above-mentioned considerations and (93), (80) and (82)

$$\begin{split} 0 &< \underline{\rho}\_{n}^{(q)} = \Gamma\_{>}^{(q)} \left( d^{(q),S} \right)^{2n} = \frac{q}{2} \cdot \left( a\_{n}^{(q),S} - \mathbf{x}\_{0}^{(q)} \right)^{2} < \frac{q}{2} \cdot \left( a\_{n}^{(q)} - \mathbf{x}\_{0}^{(q)} \right)^{2} \\ &= \frac{q}{2} \left( a\_{n}^{(q)} \right)^{2} + q \cdot \left( \mathbf{c}^{\mathbf{x}\_{0}^{(q)}} - \mathbf{x}\_{0}^{(q)} \right) \cdot a\_{n}^{(q)} + \mathbf{x}\_{0}^{(q)} \cdot \left( 1 - q \mathbf{c}^{\mathbf{x}\_{0}^{(q)}} + \frac{q}{2} \mathbf{x}\_{0}^{(q)} \right) - d^{(q),T} a\_{n}^{(q)} - \mathbf{c}^{(q),T} \\ &= \underline{\mathbf{v}}\_{\lambda}^{(q)} \left( a\_{n}^{(q)} \right) - d^{(q),T} a\_{n}^{(q)} - \mathbf{c}^{(q),T} < \boldsymbol{\xi}\_{\lambda}^{(q)} \left( a\_{n}^{(q)} \right) - d^{(q),T} a\_{n}^{(q)} - \mathbf{c}^{(q),T} \\ &< a\_{n+1}^{(q)} - d^{(q),T} \underline{a}\_{n}^{(q)} - \mathbf{c}^{(q),T} = a\_{n+1}^{(q)} - \boldsymbol{\xi}\_{\lambda}^{(q),T} \left( \underline{a}\_{n}^{(q)} \right). \end{split}$$

Hence, *a* (*q*) *<sup>n</sup>*+<sup>1</sup> < *a* (*q*) *n*+1 . For the right-hand inequality in part (i), we define the quadratic function

$$\left(\overline{\mathbf{v}}\_{\lambda}^{(q)}(\mathbf{x})\right) := \frac{q}{2} \cdot \mathbf{x}^2 + \left(1 - \frac{q}{2} \mathbf{x}\_0^{(q)} - \frac{q - \beta\_\lambda}{\mathbf{x}\_0^{(q)}}\right) \cdot \mathbf{x} + q - \beta\_\lambda \tag{A43}$$

which is a functional upper bound for *ξ* (*q*) *λ* (·) on the interval [0, *<sup>x</sup>* (*q*) 0 ] since there holds *υ* (*q*) *λ* (0) = *ξ* (*q*) *λ* (0) = *<sup>q</sup>* − *<sup>β</sup>λ*, *<sup>υ</sup>* (*q*) *λ* (*x* (*q*) 0 ) = *ξ* (*q*) *λ* (*x* (*q*) 0 ) = *x* (*q*) 0 and additionally *υ* (*q*)00 *λ* (*x*) = *q* < *qe<sup>x</sup>* = *ξ* (*q*)00 *λ* (*x*) on ]0, *x* (*q*) 0 [. Obviously, *a* (*q*) <sup>1</sup> = *q* − *β<sup>λ</sup>* = *a* (*q*) 1 . By assuming *a* (*q*) *<sup>n</sup>* ≥ *a* (*q*) *<sup>n</sup>* for *<sup>n</sup>* ∈ N, we obtain with (80), (82) and (95)

$$\begin{split} 0 &> \overline{\rho}\_{\boldsymbol{n}}^{(q)} = -\Gamma\_{\geq}^{(q)} \cdot \left( d^{(q),\mathcal{S}} \right)^{\mathfrak{t}} \cdot \left( 1 - \left( d^{(q),\mathcal{T}} \right)^{\mathfrak{n}} \right) \\ &> -\frac{q}{2} \cdot \left( \mathbf{x}\_{0} - a\_{\boldsymbol{n}}^{(q)} \right) \cdot a\_{\boldsymbol{n}}^{(q)} = \overline{\mathbf{v}}\_{\boldsymbol{\lambda}}^{(q)}(a\_{\boldsymbol{n}}^{(q)}) - \frac{\mathbf{x}\_{0}^{(q)} - (q - \beta\_{\lambda})}{\mathbf{x}\_{0}^{(q)}} \cdot a\_{\boldsymbol{n}}^{(q)} - (q - \beta\_{\lambda}) \\ &> \widetilde{\mathsf{s}}\_{\boldsymbol{\lambda}}^{(q)}(a\_{\boldsymbol{n}}^{(q)}) - d^{(q),\mathcal{S}} a\_{\boldsymbol{n}}^{(q)} - \boldsymbol{\varepsilon}^{(q),\mathcal{S}} > \widetilde{\mathsf{s}}\_{\boldsymbol{\lambda}}^{(q)}(a\_{\boldsymbol{n}}^{(q)}) - d^{(q),\mathcal{S}} \widetilde{\mathsf{a}}\_{\boldsymbol{n}}^{(q),\mathcal{S}} - \boldsymbol{\varepsilon}^{(q),\mathcal{S}} = a\_{\boldsymbol{n}+1}^{(q)} - \widetilde{\mathsf{s}}\_{\boldsymbol{\lambda}}^{(q),\mathcal{S}}(\mathbb{Z}\_{\boldsymbol{n}}^{(q)}) \,. \end{split} \tag{A44}$$

which implies *a* (*q*) *<sup>n</sup>*+<sup>1</sup> > *a* (*q*) *n*+1 . The explicit representations of the sequences *a* (*q*) *n n*∈N and *a* (*q*) *n n*∈N follow from (86) by employing the appropriate constants mentioned in the prelude of Lemma 3. By means of (83) to (85) and (86), we directly get *a* (*q*) *<sup>n</sup>* > *a* (*q*),*T <sup>n</sup>* for all *<sup>n</sup>* ∈ N, whereas *<sup>a</sup>* (*q*) *<sup>n</sup>* < *a* (*q*),*S <sup>n</sup>* holds only for all *<sup>n</sup>* ≥ <sup>2</sup>, since *<sup>ρ</sup>*<sup>0</sup> = <sup>0</sup> implies that *<sup>a</sup>* (*q*) <sup>1</sup> = *a* (*q*),*S* <sup>1</sup> = *a* (*q*) <sup>1</sup> = *<sup>q</sup>* − *<sup>β</sup>λ*.

The second part (ii) can be proved in the same way as part (ii) of (a), by employing the representation (A36). For the lower bound one has

$$\underline{a}\_{n}^{(q)} = \sum\_{k=0}^{n-1} \left( d^{(q),T} \right)^{n-1-k} \cdot \left[ c^{(q),T} + \underline{\rho}\_{k}^{(q)} \right], \qquad \text{with } c^{(q),T} > 0 \quad \text{and } \underline{\rho}\_{k}^{(q)} > 0.$$

For the upper bound we get

$$\overline{\pi}\_n^{(q)} \;= \sum\_{k=0}^{n-1} \left( d^{(q),S} \right)^{n-1-k} \cdot \left[ \mathfrak{c}^{(q),S} + \overline{\rho}\_k^{(q)} \right] \;.$$

hence it is enough to show *c* (*q*),*<sup>S</sup>* + *ρ* (*q*) *<sup>n</sup>* > 0 for all *<sup>n</sup>* ∈ N0. Considering the first two lines of calculation (A44) and incorporating *c* (*q*),*<sup>S</sup>* <sup>=</sup> *<sup>q</sup>* <sup>−</sup> *<sup>β</sup>λ*, this can be seen from

$$\left(\mathbf{c}^{(q),\mathcal{S}} + \overline{\rho}\_n^{(q)} > \overline{\mathbf{v}}\_{\lambda}^{(q)}(a\_n^{(q)}) - \frac{\mathbf{x}\_0^{(q)} - (q - \beta\_\lambda)}{\mathbf{x}\_0^{(q)}} \cdot a\_n^{(q)} \right. \\ \left. + a\_n^{(q)} \right) = \overline{\mathbf{v}}\_{\lambda}^{(q)}(a\_n^{(q)}) - d^{(q),\mathcal{S}} \cdot a\_n^{(q)} \\ > 0 \,, \quad \mathbf{c}^{(q)} = \overline{\mathbf{v}}\_{\lambda}^{(q)}(a\_n^{(q)}) - d^{(q),\mathcal{S}}$$

because on [0, *x* (*q*) 0 ] there holds *d* (*q*),*S* · *x* < *x* < *υ* (*q*) *λ* (*x*). The last part (iii) can be easily deduced from (i) together with lim*n*→<sup>∞</sup> *<sup>n</sup>* · *d* (*q*),*S n*−<sup>1</sup> = 0.

The proofs of all Theorems 5–9 are mainly based on the following

**Lemma A4.** *Recall the quantity B*e (*p*,*q*) *λ*,*X*0,*n from* (42) *for general <sup>p</sup>* ≥ 0, *<sup>q</sup>* > 0 *(notice that we do not consider parameters p* < 0*, q* ≤ 0 *in Section 6) as well as the constants d* (*q*),*T* , *d* (*q*),*S and* Γ (*q*) <sup>&</sup>lt; , Γ (*q*) <sup>&</sup>gt; *defined in* (76)*,* (77) *and* (91)*. For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P × R\{0, 1}*, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *there holds*

*(a) in the case p* ≥ 0 *and* 0 < *q* < *β<sup>λ</sup>*

$$\begin{split} \widetilde{\mathcal{B}}\_{\lambda,\mathcal{X}\_0,\mathfrak{a}}^{(p,q)} &\geq \ \exp\left\{ \mathbf{x}\_0^{(q)} \cdot \left[ \mathbf{X}\_0 - \frac{p}{q} \cdot \frac{d^{(q),T}}{1 - d^{(q),T}} \right] \cdot \left( \mathbf{1} - \left( d^{(q),T} \right)^n \right) \right. \\ &\left. + \underline{\mathcal{I}}\_n^{(q)} \cdot \mathbf{X}\_0 + \frac{p}{q} \cdot \underline{\mathfrak{a}}\_n^{(q)} \right\} =: \mathcal{C}\_{\lambda,\mathcal{X}\_0,\mathfrak{a}}^{(p,q),\mathcal{L}} \end{split} \tag{A45}$$

*B*e (*p*,*q*) *<sup>λ</sup>*,*X*0,*<sup>n</sup>* <sup>≤</sup> exp ( *x* (*q*) 0 · " *X*<sup>0</sup> − *p q* · *d* (*q*),*S* 1 − *d* (*q*),*S* # · 1 − *d* (*q*),*S n* + *p q* · *β<sup>λ</sup>* + *x* (*q*) 0 − *α<sup>λ</sup>* · *n* − *ζ* (*q*) *n* · *X*<sup>0</sup> − *p q* · *ϑ* (*q*) *n* ) =: *C* (*p*,*q*),*U λ*,*X*0,*n* , (A46)

$$\text{where} \quad \underline{\zeta}\_n^{(q)} := \Gamma\_{<}^{(q)} \cdot \frac{\left(d^{(q),T}\right)^{n-1}}{1 - d^{(q),T}} \cdot \left(1 - \left(d^{(q),T}\right)^n\right) > 0 \,, \tag{A47}$$

$$\underline{\theta}\_{n}^{(q)} \quad := \quad \Gamma\_{\leqslant}^{(q)} \cdot \frac{1 - \left(d^{(q),T}\right)^{n}}{\left(1 - d^{(q),T}\right)^{2}} \cdot \left[1 - \frac{d^{(q),T}\left(1 + \left(d^{(q),T}\right)^{n}\right)}{1 + d^{(q),T}}\right] > 0 \,,\tag{A48}$$

$$\overline{\xi}\_{n}^{(q)} \quad := \quad \Gamma\_{<}^{(q)} \cdot \left[ \frac{\left(d^{(q),S}\right)^{n} - \left(d^{(q),T}\right)^{n}}{d^{(q),S} - d^{(q),T}} - \left(d^{(q),S}\right)^{n-1} \cdot \frac{1 - \left(d^{(q),T}\right)^{n}}{1 - d^{(q),T}} \right] \\ > 0 \,, \tag{A49}$$

$$\overline{\theta}\_n^{(q)} \quad := \quad \Gamma\_{<}^{(q)} \cdot \frac{d^{(q),T}}{1 - d^{(q),T}} \cdot \left[ \frac{1 - \left(d^{(q),S}d^{(q),T}\right)^n}{1 - d^{(q),S}d^{(q),T}} - \frac{\left(d^{(q),S}\right)^n - \left(d^{(q),T}\right)^n}{d^{(q),S} - d^{(q),T}} \right] \\ > 0 \,. \tag{A50}$$

*(b) in the case p* ≥ 0 *and* 0 < *q* = *β<sup>λ</sup>*

$$\widetilde{B}\_{\lambda, X\_0, n}^{(p, q)} = \exp\left\{ \left( \frac{p}{q} \cdot \left( \beta\_\lambda + \mathfrak{x}\_0^{(q)} \right) - \mathfrak{a}\_\lambda \right) \cdot n \right\} \\ = \exp\left\{ \left( p - \mathfrak{a}\_\lambda \right) \cdot n \right\}.$$

*(c) in the case <sup>p</sup>* <sup>≥</sup> <sup>0</sup> *and* max{0 , *<sup>β</sup>λ*} <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; min <sup>n</sup> 1 , *e βλ*−1 o *the bounds C* (*p*,*q*),*L λ*,*X*0,*n and C* (*p*,*q*),*U λ*,*X*0,*n from* (96) *and* (97) *remain valid, but with*

$$\underline{\mathsf{L}}\_{n}^{(q)} := \quad \Gamma\_{>}^{(q)} \cdot \frac{\left(d^{(q),T}\right)^{n} - \left(d^{(q),S}\right)^{2n}}{d^{(q),T} - \left(d^{(q),S}\right)^{2}} > 0 \,,\tag{A51}$$

$$\underline{\mathsf{g}}\_{n}^{(q)} := \quad \frac{\Gamma\_{>}^{(q)}}{d^{(q),T} - \left(d^{(q),S}\right)^{2}} \cdot \left[\frac{d^{(q),T} \cdot \left(1 - \left(d^{(q),T}\right)^{n}\right)}{1 - d^{(q),T}} - \frac{\left(d^{(q),S}\right)^{2} \cdot \left(1 - \left(d^{(q),S}\right)^{2n}\right)}{1 - \left(d^{(q),S}\right)^{2}}\right] \\ \phantom{\tag{A52}} \tag{A52}$$

$$\overline{\zeta}\_n^{(q)} := \quad \Gamma\_{\geq}^{(q)} \cdot \left( d^{(q), \mathcal{S}} \right)^{n-1} \cdot \left[ n - \frac{1 - \left( d^{(q), \mathcal{T}} \right)^n}{1 - d^{(q), \mathcal{T}}} \right] > 0 \,, \tag{A53}$$

$$\begin{split} \overline{\theta}\_{n}^{(q)} &:= \quad \Gamma\_{>}^{(q)} \cdot \left[ \frac{d^{(q),S} - d^{(q),T}}{\left(1 - d^{(q),S}\right)^{2} \left(1 - d^{(q),T}\right)} \cdot \left(1 - \left(d^{(q),S}\right)^{n}\right)^{n} \right. \\ &\left. + \frac{d^{(q),T} \left(1 - \left(d^{(q),S}d^{(q),T}\right)^{n}\right)}{\left(1 - d^{(q),T}\right) \left(1 - d^{(q),S}d^{(q),T}\right)} - \frac{\left(d^{(q),S}\right)^{n}}{1 - d^{(q),S}} \cdot n \right]. \tag{A54} \end{split} \tag{A55}$$

*(d) for the special choices p* := *p E λ* := *α λ* A *α* 1−*λ* <sup>H</sup> <sup>&</sup>gt; 0, *<sup>q</sup>* :<sup>=</sup> *<sup>q</sup> E λ* := *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; <sup>0</sup> *in the parameter setup* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) ∈ (P*NI* ∪ P*SP,1*)× ]*λ*−, *<sup>λ</sup>*+[ \{0, 1} *we obtain*

$$\lim\_{n \to \infty} \frac{1}{n} \log \left( V\_{\lambda, \mathbf{X}\_0, \mu} \right) \\ = \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathbf{C}\_{\lambda, \mathbf{X}\_0, \mu}^{(p\_\lambda^\mathbb{F}, q\_\lambda^\mathbb{F}), L} \right) \\ = \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathbf{C}\_{\lambda, \mathbf{X}\_0, \mu}^{(p\_\lambda^\mathbb{F}, q\_\lambda^\mathbb{F}), L} \right) \\ = \frac{\mathfrak{a}\_{\mathcal{A}}}{\mathcal{P}\_{\mathcal{A}}} \cdot \mathbf{x}\_0^{(q\_\lambda^\mathbb{F})} \cdot \mathbf{x}\_0^{(p\_\lambda^\mathbb{F})} $$

*(e) for all general p* <sup>≥</sup> <sup>0</sup> *with either* <sup>0</sup> <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; *<sup>β</sup><sup>λ</sup> or* max{0, *<sup>β</sup>λ*} <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; min <sup>n</sup> 1 , *e βλ*−1 o *we get*

$$\lim\_{n \to \infty} \frac{1}{n} \log \left( \hat{B}^{(p,q)}\_{\lambda, \mathbf{X}\_0, \mu} \right) \\ = \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{C}^{(p,q),L}\_{\lambda, \mathbf{X}\_0, \mu} \right) \\ = \lim\_{n \to \infty} \frac{1}{n} \log \left( \mathcal{C}^{(p,q),II}\_{\lambda, \mathbf{X}\_0, \mu} \right) \\ = \frac{p}{q} \cdot \left( \beta\_\lambda + \mathbf{x}\_0^{(q)} \right) - \alpha\_\lambda \cdot \mathbf{x}\_0^{(p,q)}$$

**Proof of Lemma A4.** The closed-form bounds *C* (*p*,*q*),*L λ*,*X*0,*n* and *C* (*p*,*q*),*U λ*,*X*0,*n* are obtained by substituting in the representation (42) (for *<sup>B</sup>*<sup>e</sup> (*p*,*q*) *λ*,*X*0,*n* , cf. Theorem 1) the recursive sequence member *a* (*q*) *<sup>n</sup>* by the explicit sequence member *a* (*q*) *<sup>n</sup>* respectively *a* (*q*) *<sup>n</sup>* . From the definitions of these sequences (92) to (95) and from (83) to (85) one can see that we basically have to evaluate the term

$$\exp\left\{ \left( \hat{a}\_n^{\text{hom}} + \widetilde{c}\_n \right) \cdot X\_0 \right.\\ \left. + \left. \frac{p}{q} \cdot \sum\_{k=1}^n \left( \hat{a}\_k^{\text{hom}} + \widetilde{c}\_k \right) \right.\\ \left. + \left. \left( \frac{p}{q} \cdot \beta\_\lambda - a\_\lambda \right) \cdot n \right. \right\}, \tag{A55}$$

where <sup>e</sup>*<sup>a</sup> hom <sup>n</sup>* <sup>+</sup> *<sup>c</sup>*e*<sup>n</sup>* <sup>=</sup> <sup>e</sup>*a<sup>n</sup>* is either interpreted as the lower approximate *<sup>a</sup>* (*q*) *<sup>n</sup>* or as the upper approximate *a* (*q*) *<sup>n</sup>* . After rearranging and incorporating that *<sup>c</sup>* (*q*),*S* 1−*d* (*q*),*<sup>S</sup>* = *<sup>c</sup>* (*q*),*T* 1−*d* (*q*),*<sup>T</sup>* = *x* (*q*) 0 in both approximate cases, we obtain with the help of (86), (87) for the expression (A55) in the case <sup>0</sup> ≤ *<sup>ν</sup>* < κ < *<sup>d</sup>*

$$\begin{split} & \exp\left\{ \mathbf{x}\_{0}^{(q)} \cdot (\mathbf{1} - d^{n}) \cdot \left[ \mathbf{X}\_{0} - \frac{p}{q} \cdot \frac{d}{\mathbf{1} - d} \right] + \left( \frac{p}{q} \cdot \left( \beta\_{\lambda} + \mathbf{x}\_{0}^{(q)} \right) - a\_{\lambda} \right) \cdot \mathbf{n} \right. \\ & \left. + \left[ \mathbf{K}\_{1} \cdot \frac{d^{n} - \varkappa^{n}}{d - \varkappa} + \mathbf{K}\_{2} \cdot \frac{d^{n} - \nu^{n}}{d - \nu} \right] \cdot \mathbf{X}\_{0} \\ & \left. + \frac{p}{q} \cdot \left[ \left( \frac{\mathbf{K}\_{1}}{d - \varkappa} + \frac{\mathbf{K}\_{2}}{d - \nu} \right) \cdot \frac{d \cdot (1 - d^{n})}{1 - d} - \frac{\mathbf{K}\_{1} \cdot \varkappa \cdot (1 - \varkappa^{n})}{(d - \varkappa)(1 - \varkappa)} - \frac{\mathbf{K}\_{2} \cdot \nu \cdot (1 - \nu^{n})}{(d - \nu)(1 - \nu)} \right] \right\}. \end{split} \tag{A56}$$

In the other case <sup>0</sup> ≤ *<sup>ν</sup>* < κ = *<sup>d</sup>*, the application of (88), (89) turns (A55) into

$$\begin{split} & \exp\left\{ \mathbf{x}\_{0}^{(q)} \cdot (1 - d^{n}) \cdot \left[ \mathbf{X}\_{0} - \frac{p}{q} \cdot \frac{d}{1 - d} \right] + \left( \frac{p}{q} \cdot \left( \beta\_{\lambda} + \mathbf{x}\_{0}^{(q)} \right) - \mathbf{a}\_{\lambda} \right) \cdot n \right. \\ & \left. + \left[ \mathbf{K}\_{1} \cdot n \cdot d^{n - 1} + \mathbf{K}\_{2} \cdot \frac{d^{n} - \nu^{n}}{d - \nu} \right] \cdot \mathbf{X}\_{0} \\ & \left. + \frac{p}{q} \cdot \left[ \left( \frac{\mathbf{K}\_{1}}{d(1 - d)} + \frac{\mathbf{K}\_{2}}{d - \nu} \right) \cdot \frac{d \cdot (1 - d^{n})}{1 - d} - \frac{\mathbf{K}\_{2} \cdot \nu \cdot (1 - \nu^{n})}{(d - \nu)(1 - \nu)} - \frac{\mathbf{K}\_{1} \cdot d^{n}}{1 - d} \cdot n \right] \right\}. \end{split} \tag{A57}$$

After these preparatory considerations let us now begin with elaboration of the details.

(a) Let 0 < *<sup>q</sup>* < *<sup>β</sup>λ*. We obtain a closed-form lower bound for *<sup>B</sup>*<sup>e</sup> (*p*,*q*) *λ*,*X*0,*n* by employing the parameters *<sup>c</sup>* <sup>=</sup><sup>b</sup> *<sup>c</sup>* (*q*),*T* , *<sup>d</sup>* <sup>=</sup><sup>b</sup> *<sup>d</sup>* (*q*),*T* , *K*<sup>2</sup> = *ν* = 0, *K*<sup>1</sup> = Γ (*q*) <sup>&</sup>lt; , and κ = *d* (*q*),*T* 2 , cf. (93) in combination with (85). Since κ < *d* (*q*),*T* , we have to plug in these parameters into (A56). The representations of *ζ* (*q*) *n* and *ϑ* (*q*) *n* in (A47) and (A48) follow immediately. For a closed-form upper bound, we employ the parameters *<sup>c</sup>* <sup>=</sup><sup>b</sup> *<sup>c</sup>* (*q*),*S* , *<sup>d</sup>* <sup>=</sup><sup>b</sup> *<sup>d</sup>* (*q*),*S* , −*K*<sup>1</sup> = *K*<sup>2</sup> = Γ (*q*) <sup>&</sup>lt; , κ = *d* (*q*),*<sup>T</sup>* and *ν* = *d* (*q*),*Sd* (*q*),*T* (in particular, κ < *d* (*q*),*S* implying that

we have to use (A56)). From this, (A49) can be deduced directly; the representation (A50) comes from the expressions in the squared brackets in the last line of (A56) and from

− Γ (*q*) < *d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* − Γ (*q*) < *d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*Sd* (*q*),*T* ! · *d* (*q*),*S* · 1 − *d* (*q*),*S n* 1 − *d* (*q*),*S* + Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*T* · 1 − *d* (*q*),*T n d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* <sup>1</sup> <sup>−</sup> *<sup>d</sup>* (*q*),*T* − Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*Sd* (*q*),*T* · 1 − *d* (*q*),*Sd* (*q*),*T n d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*Sd* (*q*),*T* <sup>1</sup> <sup>−</sup> *<sup>d</sup>* (*q*),*Sd* (*q*),*T* = − Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*T* 1 − *d* (*q*),*S d* (*q*),*S d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* <sup>1</sup> <sup>−</sup> *<sup>d</sup>* (*q*),*T* · *d* (*q*),*S* · 1 − *d* (*q*),*S n* 1 − *d* (*q*),*S* + Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*T* · 1 − *d* (*q*),*T n d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* <sup>1</sup> <sup>−</sup> *<sup>d</sup>* (*q*),*T* − Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*T* · 1 − *d* (*q*),*Sd* (*q*),*T n* 1 − *d* (*q*),*T* <sup>1</sup> <sup>−</sup> *<sup>d</sup>* (*q*),*Sd* (*q*),*T* = − Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*T* 1 − *d* (*q*),*T* · 1 − *d* (*q*),*Sd* (*q*),*T n* 1 − *d* (*q*),*Sd* (*q*),*T* + 1 − *d* (*q*),*S n d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* − 1 − *d* (*q*),*T n d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* = − Γ (*q*) <sup>&</sup>lt; · *d* (*q*),*T* 1 − *d* (*q*),*T* · 1 − *d* (*q*),*Sd* (*q*),*T n* 1 − *d* (*q*),*Sd* (*q*),*T* − *d* (*q*),*S n* − *d* (*q*),*T n d* (*q*),*<sup>S</sup>* <sup>−</sup> *<sup>d</sup>* (*q*),*T* <sup>=</sup> <sup>−</sup>*<sup>ϑ</sup>* (*q*) *n* .

Part (b) has already been mentioned in Remark 1 (b) and is due to the fact that for 0 < *q* = *βλ*, the sequence *a* (*q*) *n n*∈N is itself explicitly representable by *a* (*q*) *<sup>n</sup>* = 0 for all *<sup>n</sup>* ∈ N (cf. Properties 1 (P2)). Plugging this into (42) gives the desired result.

(c) Let us now consider max{0, *<sup>β</sup>λ*} < *<sup>q</sup>* < min{1,*<sup>e</sup> <sup>β</sup>λ*−1}. For a closed-form lower bound for *<sup>B</sup>*<sup>e</sup> (*p*,*q*) *<sup>λ</sup>*,*X*0,*<sup>n</sup>* we have to employ the parameters *<sup>c</sup>* <sup>=</sup><sup>b</sup> *<sup>c</sup>* (*q*),*T* , *<sup>d</sup>* <sup>=</sup><sup>b</sup> *<sup>d</sup>* (*q*),*T* , *K*<sup>2</sup> = *ν* = 0, *K*<sup>1</sup> = Γ (*q*) <sup>&</sup>gt; and κ = *d* (*q*),*S* 2 , cf. (93) in combination with (85). The representations of *ζ* (*q*) *n* and *ϑ* (*q*) *<sup>n</sup>* in (A51) and (A52) follow immediately from (A56). For a closed-form upper bound, we use the parameters *<sup>c</sup>* <sup>=</sup><sup>b</sup> *<sup>c</sup>* (*q*),*S* , *<sup>d</sup>* <sup>=</sup><sup>b</sup> *<sup>d</sup>* (*q*),*S* , −*K*<sup>1</sup> = *<sup>K</sup>*<sup>2</sup> = <sup>Γ</sup> (*q*) <sup>&</sup>gt; , κ = *d* (*q*),*<sup>S</sup>* and *ν* = *d* (*q*),*Sd* (*q*),*T* . Notice that in this case we stick to the representation (A57). The formula (104) is obviously valid, and (105) is implied by

$$\begin{split} & \left( \frac{-\Gamma\_{>}^{(q)}}{d^{(q),S}\left(1-d^{(q),S}\right)} + \frac{\Gamma\_{>}^{(q)}}{d^{(q),S}-d^{(q),S}d^{(q),T}} \right) \cdot \frac{d^{(q),S}\cdot\left(1-\left(d^{(q),S}\right)^{\eta}\right)}{1-d^{(q),S}} \\ &= \quad -\Gamma\_{>}^{(q)} \cdot \frac{d^{(q),S}-d^{(q),T}}{\left(1-d^{(q),S}\right)^{2}\left(1-d^{(q),T}\right)} \cdot \left(1-\left(d^{(q),S}\right)^{\eta}\right) \ . \end{split}$$

The parts (d) and (e) are trivial by incorporating that in all respective cases one has *d* (*q*),*<sup>S</sup>* <sup>∈</sup>]0, 1[, *d* (*q*),*<sup>T</sup>* <sup>∈</sup>]0, 1[ and lim*n*→<sup>∞</sup> *<sup>n</sup>* · *<sup>d</sup>* (*q*),*<sup>S</sup>* = 0.

**Proof of Theorem 5.** (a) For *<sup>λ</sup>* ∈ ]0, 1[, we get 0 < *<sup>q</sup> E <sup>λ</sup>* < *<sup>β</sup><sup>λ</sup>* and the assertion follows by applying part (a) of Lemma A4. Notice that in the current subcase <sup>P</sup>NI ∪ PSP,1 there holds *<sup>p</sup> E λ q E λ <sup>β</sup><sup>λ</sup>* − *<sup>α</sup><sup>λ</sup>* = 0 as well as *p E λ q E λ* = *α*A *β*A = *α*H *β*H . For the case *<sup>λ</sup>* ∈ R\[0, 1], one gets from Lemma A1 that max{0, *<sup>β</sup>λ*} < *<sup>q</sup> E λ* , and there holds *q E <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1} iff *<sup>λ</sup>* <sup>∈</sup>]*λ*−, *<sup>λ</sup>*+[ \[0, 1], cf. Lemma 1. Thus, an application of part (c) of Lemma A4 proves the desired result. The assertion (b) is equivalent to part (d) of Lemma A4.

**Proof of Theorem 6.** The assertions follow immediately from (A45), Lemma A4(b),(e), Proposition 6(d) as well as the incorporation of the fact that for *<sup>λ</sup>* ∈]0, 1[ there holds *<sup>q</sup> L <sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>lt; *<sup>β</sup><sup>λ</sup>* in the case (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) <sup>∈</sup> (PSP\(PSP,1 ∪ PSP,4)) (i.e., *<sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*H) respectively *<sup>q</sup> L <sup>λ</sup>* = *<sup>β</sup><sup>λ</sup>* in the case (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP,4 (i.e., *<sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*H).

**Proof of Theorem 7.** This can be deduced from (A46), from the parts (b), (c) and (e) of Lemma A4 as well as the incorporation of *p U <sup>λ</sup>* ≥ *α λ* A *α* 1−*λ* <sup>H</sup> <sup>&</sup>gt; <sup>0</sup> for *<sup>λ</sup>* <sup>∈</sup>]0, 1[. Notice that an inadequate choice of *<sup>p</sup> U λ* , *q U λ* may lead to *<sup>p</sup> U λ q U λ* (*β<sup>λ</sup>* + *x* (*q U λ* ) 0 ) − *<sup>α</sup><sup>λ</sup>* > <sup>0</sup>.

**Proof of Theorem 8.** The assertions follow immediately from (A45) and from the parts (b), (c) and (e) of Lemma A4. Notice that an inadequate choice of *p L λ* , *q L <sup>λ</sup>* may lead to *<sup>p</sup> L λ q L λ* (*β<sup>λ</sup>* + *x* (*q U λ* ) 0 ) − *<sup>α</sup><sup>λ</sup>* < <sup>0</sup>.

**Proof of Theorem 9.** Let *p U <sup>λ</sup>* = *α λ* A *α* 1−*λ* <sup>H</sup> <sup>&</sup>gt; max{0, *<sup>α</sup>λ*} and *<sup>q</sup> U <sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; max{0, *<sup>β</sup>λ*}. Since *<sup>q</sup> U <sup>λ</sup>* < min{1,*e <sup>β</sup>λ*−1} iff *<sup>λ</sup>* <sup>∈</sup>]*λ*−, *<sup>λ</sup>*+[ \[0, 1] (cf. Lemma 1 for *<sup>q</sup><sup>λ</sup>* :<sup>=</sup> *<sup>q</sup> U λ* )), this theorem follows from (A46) of Lemma A4, from the parts (b), (e) of Lemma A4 and from part (d) of Proposition 14.

#### *Appendix A.4. Proofs and Auxiliary Lemmas for Section 7*

**Proof of Theorem 10.** As already mentioned above, one can adapt the proof of Theorem 9.1.3 in Ethier & Kurtz [138] who deal with drift-parameters *<sup>η</sup>* = <sup>0</sup>, *<sup>κ</sup>*• = <sup>0</sup>, and the different setup of *<sup>σ</sup>*−*independent time-scale* and a sequence of *critical* Galton-Watson processes *without immigration* with *general* offspring distribution. For the sake of brevity, we basically outline here only the main differences to their proof; for similar limit investigations involving offspring/immigration distributions and parametrizations which are incompatble to ours, see e.g., Sriram [142].

As a first step, let us define the generator

$$A\bullet f(\mathbf{x}) := \left(\eta - \kappa\_{\bullet} \cdot \mathbf{x}\right) \cdot f'(\mathbf{x}) + \frac{\sigma^2}{2} \cdot \mathbf{x} \cdot f''(\mathbf{x}), \quad f \in \mathsf{C}\_c^\infty\left([0, \infty)\right), \dots$$

which corresponds to the diffusion process *X*e governed by (133). In connection with (130), we study

$$\mathbb{E}\_{\bullet}^{(m)} f(\mathbf{x}) := \mathbb{E} P\_{\bullet} \left[ f \left( \frac{1}{m} \left( \sum\_{k=1}^{m\infty} \mathsf{Y}\_{0,k}^{(m)} + \hat{\mathsf{Y}}\_{0}^{(m)} \right) \right) \right], \quad \mathbf{x} \in E^{(m)} := \frac{1}{m} \mathbb{N}\_{0}, \quad f \in \mathsf{C}\_{\varepsilon}^{\infty} ([0, \infty), \epsilon)$$

where the *Y* (*m*) 0,*k* , *Y*e (*m*) 0 are independent and (Poisson-*β* (*m*) • respectively Poisson-*<sup>α</sup>* (*m*) • ) distributed as the members of the collection *Y* (*m*) respectively *<sup>Y</sup>*e(*m*) . By the Theorems 8.2.1 and 1.6.5 as well as Corollary 4.8.9 of [138] it is sufficient to show

$$\lim\_{m \to \infty} \sup\_{\mathbf{x} \in E^{(m)}} \left| \sigma^2 m \left( T\_{\bullet}^{(m)} f(\mathbf{x}) - f(\mathbf{x}) \right) - A\_{\bullet} f(\mathbf{x}) \right| \\ = 0, \ f \in \mathsf{C}\_c^{\infty} \left( [0, \infty) \right). \tag{A58}$$

But (A58) follows mainly from the next

**Lemma A5.** *Let*

$$S\_n^{(m)} := \frac{1}{\sqrt{n!}} \left( \sum\_{k=1}^n \left( Y\_{0,k}^{(m)} - \beta\_\bullet^{(m)} \right) + \hat{Y}\_0^{(m)} - a\_\bullet^{(m)} \right), \quad n \in \mathbb{N}, \ m \in \overline{\mathbb{N}}\_+$$

*with the usual convention S*(*m*) 0 := 0*. Then for all m* ∈ N*, x* ∈ *E* (*m*) *and all f* ∈ *C* ∞ *c* [0, ∞) 

$$\mathbf{e}^{(m)}(\mathbf{x}) := \operatorname{EP}\_{\bullet} \left[ \int\_{0}^{1} \left( \mathbf{S}\_{\text{mx}}^{(m)} \right)^{2} \mathbf{x} (1 - \mathbf{z}) \left( f^{\prime \prime} \left( \mathbf{\mathcal{S}}\_{\bullet}^{(m)} \mathbf{x} + \frac{\mathbf{a}\_{\bullet}^{(m)}}{m} + \nu \sqrt{\frac{\mathbf{x}}{m}} \mathbf{S}\_{\text{mx}}^{(m)} \right) - f^{\prime \prime}(\mathbf{x}) \right) d\nu \right]$$

$$= \frac{1}{\sigma^{2}} \cdot \left[ \sigma^{2} m \cdot \left( T\_{\bullet}^{(m)} f(\mathbf{x}) - f(\mathbf{x}) \right) - A\_{\bullet} f(\mathbf{x}) \right] \\ + \mathcal{R}^{(m)}, \qquad \text{where } \lim\_{m \to \infty} \mathcal{R}^{(m)} = 0. \tag{A59}$$

**Proof of Lemma A5.** Let us fix *<sup>f</sup>* ∈ *<sup>C</sup>* ∞ *c* [0, ∞) . From the involved Poissonian expectations it is easy to see that

$$\lim\_{m \to \infty} \left| \sigma^2 m \left( T^{(m)}\_{\bullet} f(0) - f(0) \right) - A\_{\bullet} f(0) \right|\_{\cdot} = 0 \text{ \AA}$$

and thus (A59) holds for *<sup>x</sup>* = <sup>0</sup>. Accordingly, we next consider the case *<sup>x</sup>* ∈ *<sup>E</sup>* (*m*)\{0}, with fixed *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>. From *EP*• *S* (*m*) *mx* <sup>2</sup> = *β* (*m*) • + *α* (*m*) • *mx* we obtain

$$EP\_{\bullet}\left[\left(S\_{\mathrm{mx}}^{(m)}\right)^{2}\mathbf{x}f''(\mathbf{x})\int\_{0}^{1}(1-v)dv\right] \\ \quad = \frac{1}{2}\left(\mathcal{J}\_{\bullet}^{(m)}\cdot\mathbf{x} + \frac{a\_{\bullet}^{(m)}}{m}\right)f''(\mathbf{x}) \\ \quad =: a\_{\mathrm{mx}}\frac{f''(\mathbf{x})}{2} \\ \quad =: a\frac{f''(\mathbf{x})}{2} .\tag{A60}$$

Furthermore, with *bmx* := *b* := *a* + √ *x*/*m* · *S* (*m*) *mx* = <sup>1</sup> *m* ∑ *mx k*=1 *Y* (*m*) 0,*<sup>k</sup>* + *Y*e (*m*) 0 we get on {*<sup>S</sup>* (*m*) *mx* 6= 0}

$$\int\_0^1 f''\left(\mathcal{S}^{(m)}\_{\bullet} \mathbf{x} + \frac{a^{(m)}\_{\bullet}}{m} + \upsilon \sqrt{\frac{\mathbf{x}}{m}} \mathcal{S}^{(m)}\_{\mathbf{mx}}\right) d\upsilon = \sqrt{\frac{m}{\mathbf{x}}} \cdot \frac{1}{\mathcal{S}^{(m)}\_{\mathbf{mx}}} \int\_a^b f''(y) dy = \sqrt{\frac{m}{\mathbf{x}}} \cdot \frac{f'(b) - f'(a)}{\mathcal{S}^{(m)}\_{\mathbf{mx}}} \tag{A61}$$

as well as

$$\int\_{0}^{1} \upsilon f^{\prime\prime} \left( \mathcal{S}^{(m)}\_{\bullet} x + \frac{a^{(m)}\_{\bullet}}{m} + \upsilon \sqrt{\frac{\chi}{m}} \, \mathcal{S}^{(m)}\_{\mathrm{mx}} \right) d\upsilon = \frac{m}{\mathrm{x} \left( \mathcal{S}^{(m)}\_{\mathrm{mx}} \right)^{2}} \left[ \int\_{a}^{b} y f^{\prime\prime}(y) \, dy - a \int\_{a}^{b} f^{\prime\prime}(y) \, dy \right]$$

$$= \sqrt{\frac{m}{\mathcal{X}}} \cdot \frac{f^{\prime}(b)}{\mathcal{S}^{(m)}\_{\mathrm{mx}}} + \frac{m}{\mathcal{X}} \cdot \frac{f(a) - f(b)}{\left( \mathcal{S}^{(m)}\_{\mathrm{mx}} \right)^{2}} \,. \tag{A62}$$

With our choice *β* (*m*) • = 1 − *κ*• *σ* <sup>2</sup>*m* and *α* (*m*) • = *β* (*m*) • · *η σ* 2 , a Taylor expansion of *f* at *x* gives

$$f(a) = f(\mathbf{x}) + \frac{1}{\sigma^2 m} \cdot f'(\mathbf{x}) \left( \mathbb{\beta}^{(m)}\_{\bullet} \cdot \eta - \mathbf{x}\_{\bullet} \cdot \mathbf{x} \right) + o\left(\frac{1}{m}\right),\tag{A63}$$

where for the case *η* = *κ* = 0 we use the convention *o* 1 *m* ≡ 0. Combining (A60) to (A63) and the centering *EP*• h *S* (*m*) *mx* <sup>i</sup> = 0, the left hand side of Equation (A59) becomes

*EP*• " Z 1 0 *S* (*m*) *mx* <sup>2</sup> *x*(1 − *v*) *f* 00 *β* (*m*) • *x* + *α* (*m*) • *m* + *v* r *x m S* (*m*) *mx* ! − *f* 00(*x*) ! *dv*# = *EP*• h√ *mx* · *S* (*m*) *mx* · *f* 0 (*b*) − *f* 0 (*a*) i <sup>−</sup> *EP*• h√ *mx* · *S* (*m*) *mx* · *f* 0 (*b*) + *<sup>m</sup>* · (*f*(*a*) <sup>−</sup> *<sup>f</sup>*(*b*))<sup>i</sup> − 1 2 *β* (*m*) • · *x* + *α* (*m*) • *m* ! · *f* 00(*x*) = *m* · *EP*• h *f*(*b*) i − *f*(*a*) − 1 2 *β* (*m*) • · *x* + *α* (*m*) • *m* ! · *f* 00(*x*) = *m* · ( *EP*• " *f* 1 *m mx* ∑ *k*=1 *Y* (*m*) 0,*<sup>k</sup>* + *Y*e 0 !!# − *f*(*x*) ) − 1 *σ* 2 *A*• *f*(*x*) + 1 *σ* 2 h (*η* − *κ*• · *x*) − *β* (*m*) • · *η* + *κ*• · *x* i · *f* 0 (*x*) + *<sup>x</sup>* 2 " 1 − *β* (*m*) • − *α* (*m*) • *m* # · *f* <sup>00</sup>(*x*) − *m* · *o* 1 *m* 

which immediately leads to the right hand side of (A59).

To proceed with the proof of Theorem 10, we obtain for *<sup>m</sup>* ≥ <sup>2</sup>*κ*•/*<sup>σ</sup>* 2 the inequality *β* (*m*) • ≥ 1/2 and accordingly for all *<sup>v</sup>* ∈]0, 1[, *<sup>x</sup>* ∈ *<sup>E</sup>* (*m*)

$$\mathcal{A}^{(m)}\_{\bullet} \ge + \frac{a^{(m)}\_{\bullet}}{m} + \upsilon \sqrt{\frac{\chi}{m}} \, S^{(m)}\_{\mathrm{mx}} = (1 - \upsilon) \cdot \mathbf{x} \cdot \boldsymbol{\mathcal{J}}^{(m)}\_{\bullet} + (1 - \upsilon) \frac{a^{(m)}\_{\bullet}}{m} + \upsilon \left( \sum\_{k=1}^{\mathrm{mx}} \mathbf{Y}^{(m)}\_{0,k} + \widetilde{\mathbf{Y}}\_{0} \right) \ge \mathbf{x} \cdot \frac{1 - \upsilon}{2} \cdot \mathbf{x} \cdot \boldsymbol{\mathcal{J}}^{(m)}\_{\bullet} + \upsilon \left( \sum\_{k=1}^{\mathrm{mx}} \mathbf{Y}^{(m)}\_{0,k} + \widetilde{\mathbf{Y}}\_{0} \right) \ge \mathbf{0}, \quad \forall \mathbf{x} \in \mathcal{X}\_{\bullet}$$

Suppose that the support of *<sup>f</sup>* is contained in the interval [0, *<sup>c</sup>*]. Correspondingly, for *<sup>v</sup>* ≤ 1 − 2*c*/*<sup>x</sup>* the integrand in *e* (*m*) (*x*) is zero and hence with (A64) we obtain the bounds

$$\begin{aligned} & \left| \int\_0^1 \left( \mathcal{S}\_{mx}^{(m)} \right)^2 \mathbf{x} (1 - \boldsymbol{\upsilon}) \left( f^{\prime\prime} \left( \mathcal{S}\_{\bullet}^{(m)} \mathbf{x} + \frac{\mathbf{a}\_{\bullet}^{(m)}}{m} + \boldsymbol{\upsilon} \sqrt{\frac{\mathbf{x}}{m}} \, \mathcal{S}\_{mx}^{(m)} \right) - f^{\prime\prime} (\boldsymbol{\upsilon}) \right) d\boldsymbol{\upsilon} \right| \\ & \leq \quad \int\_{0 \vee (1 - 2c/\boldsymbol{x})}^1 \left( \mathcal{S}\_{mx}^{(m)} \right)^2 \mathbf{x} (1 - \boldsymbol{\upsilon}) \cdot 2 \left\| f^{\prime\prime} \right\|\_{\infty} d\boldsymbol{\upsilon} \leq \left. \mathbf{x} \cdot \left( \mathcal{S}\_{mx}^{(m)} \right)^2 \left( 1 \wedge \frac{2c}{\boldsymbol{x}} \right)^2 \left\| f^{\prime\prime} \right\|\_{\infty} . \end{aligned}$$

From this, one can deduce lim*m*→<sup>∞</sup> sup*x*∈*E*(*m*) *<sup>e</sup>* (*m*) (*x*) = 0–and thus (A58) – in the same manner as at the end of the proof of Theorem 9.1.3 in [138] (by means of the dominated convergence theorem).

**Proof of Proposition 15.** Let (*κ*A, *<sup>κ</sup>*H, *<sup>η</sup>*) <sup>∈</sup> <sup>P</sup>e*NI* <sup>∪</sup> <sup>P</sup><sup>e</sup> *SP*,1 be fixed. We have to find those orders *<sup>λ</sup>* ∈ R\[0, 1] which satisfy for all sufficiently large *<sup>m</sup>* ∈ N

$$q\_{\lambda}^{(m)} = \left(1 - \frac{\kappa\_{\mathcal{A}}}{\sigma^2 m}\right)^{\lambda} \left(1 - \frac{\kappa\_{\mathcal{H}}}{\sigma^2 m}\right)^{1-\lambda} < \min\left\{1, \,\,\sigma^{\beta\_{\lambda}^{(m)} - 1}\right\}.\tag{A64}$$

In order to achieve this, we interpret *q* (*m*) *<sup>λ</sup>* = *q<sup>λ</sup>* 1 *m* in terms of the function

$$q\_{\lambda}(\mathbf{x}) := \left(1 - \frac{\kappa\_{\mathcal{A}}}{\sigma^2} \cdot \mathbf{x}\right)^{\lambda} \left(1 - \frac{\kappa\_{\mathcal{Y}}}{\sigma^2} \cdot \mathbf{x}\right)^{1-\lambda}, \qquad \mathbf{x} \in \left]-\epsilon, \epsilon\right[\tag{A65}$$

for some small enough *e* > 0 such that (A65) is well-defined. Since *β* (*m*) *<sup>λ</sup>* − 1 = − *κλ σ* 2 ·*m* = − *κλ σ* 2 · *x* = − *λκ*A+(1−*λ*)*κ*<sup>H</sup> *σ* 2 · *<sup>x</sup>*, for the verification of (A64) it suffices to show

$$\lim\_{\lambda \nearrow 0} \frac{1 - q\_\lambda(\mathbf{x})}{\mathbf{x}} \quad > \quad \mathbf{0} \, , \tag{\text{A66}}$$

$$\text{and} \qquad \lim\_{\mathbf{x} \searrow 0} \frac{e^{-\frac{\mathbf{x}\_{\lambda}}{\sigma^{2}} \cdot \mathbf{x}} - q\_{\lambda}(\mathbf{x})}{\mathbf{x}^{2}} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \tag{A67}$$

By l'Hospital's rule, one gets lim*x*&<sup>0</sup> 1−*qλ*(*x*) *<sup>x</sup>* = *λκ*A+(1−*λ*)*κ*<sup>H</sup> *σ* <sup>2</sup> = *κλ σ* <sup>2</sup> and hence

$$\text{(A66)} \iff \begin{cases} \quad \lambda < \frac{\kappa\_{\mathcal{H}}}{\kappa\_{\mathcal{H}} - \kappa\_{\mathcal{A}'}} & \text{if} \quad \kappa\_{\mathcal{A}} < \kappa\_{\mathcal{H}'} \\\\ \quad \lambda > - \frac{\kappa\_{\mathcal{H}}}{\kappa\_{\mathcal{A}} - \kappa\_{\mathcal{H}'}} & \text{if} \quad \kappa\_{\mathcal{A}} > \kappa\_{\mathcal{H}} . \end{cases} \tag{A68}$$

To find a condition that guarantees (A67), we use l'Hospital's rule twice to deduce

$$\lim\_{\mathbf{x}\searrow 0} \frac{e^{-\frac{\mathbf{x}\lambda}{\sigma^2}\cdot\mathbf{x}} - q\_{\lambda}(\mathbf{x})}{\mathbf{x}^2} = \frac{1}{2\sigma^4} \left[\kappa\_{\lambda}^2 - \lambda(\lambda - 1)(\mathbf{x}\_{\mathcal{A}} - \kappa\_{\mathcal{H}})^2\right] \\ = \frac{1}{2\sigma^4} \left[\lambda\kappa\_{\mathcal{A}}^2 + (1 - \lambda)\kappa\_{\mathcal{H}}^2\right]$$

and hence we obtain

$$\text{(A67)} \iff \begin{cases} \quad \lambda < \frac{\kappa\_{\mathcal{H}}^2}{\kappa\_{\mathcal{H}}^2 - \kappa\_{\mathcal{A}}^2}, & \text{if} \quad \kappa\_{\mathcal{A}} < \kappa\_{\mathcal{H}}, \\\\ \quad \lambda > -\frac{\kappa\_{\mathcal{H}}^2}{\kappa\_{\mathcal{A}}^2 - \kappa\_{\mathcal{H}}^2}, & \text{if} \quad \kappa\_{\mathcal{A}} > \kappa\_{\mathcal{H}}. \end{cases} \tag{A69}$$

To compare both the lower and upper bounds in (A68) and (A69), let us calculate

$$\frac{\kappa\_{\mathcal{H}}^{2}}{\kappa\_{\mathcal{H}}^{2}-\kappa\_{\mathcal{A}}^{2}}-\frac{\kappa\_{\mathcal{H}}}{\kappa\_{\mathcal{H}}-\kappa\_{\mathcal{A}}} = -\frac{\kappa\_{\mathcal{A}}\kappa\_{\mathcal{H}}}{(\kappa\_{\mathcal{H}}-\kappa\_{\mathcal{A}})(\kappa\_{\mathcal{H}}+\kappa\_{\mathcal{A}})} \left\{ < \ 0, \quad \text{if} \quad \kappa\_{\mathcal{A}} < \kappa\_{\mathcal{H}}.\tag{A70}$$

Incorporating this, we observe that both conditions (A66) and (A67) are satisfied simultaneously iff

$$\begin{array}{rclcrcl}\lambda & & & \min\left\{\frac{\kappa\_{\mathcal{H}}}{\kappa\_{\mathcal{H}} - \kappa\_{\mathcal{A}}}, \frac{\kappa\_{\mathcal{H}}^{2}}{\kappa\_{\mathcal{H}}^{2} - \kappa\_{\mathcal{A}}^{2}}\right\} & & & = & \frac{\kappa\_{\mathcal{H}}^{2}}{\kappa\_{\mathcal{H}}^{2} - \kappa\_{\mathcal{A}}^{2}} & \text{if} & \kappa\_{\mathcal{A}} < \kappa\_{\mathcal{H}},\\\lambda & & & \max\left\{-\frac{\kappa\_{\mathcal{H}}}{\kappa\_{\mathcal{A}} - \kappa\_{\mathcal{H}}}, \frac{-\kappa\_{\mathcal{H}}^{2}}{\kappa\_{\mathcal{A}}^{2} - \kappa\_{\mathcal{H}}^{2}}\right\} & & = & -\frac{\kappa\_{\mathcal{H}}^{2}}{\kappa\_{\mathcal{A}}^{2} - \kappa\_{\mathcal{H}}^{2}} & \text{if} & \kappa\_{\mathcal{A}} > \kappa\_{\mathcal{H}}.\end{array}$$

which finishes the proof.

The following lemma is the main tool for the proof of Theorem 11.

**Lemma A6.** *Let* (*κ*A, *<sup>κ</sup>*H, *<sup>η</sup>*, *<sup>λ</sup>*) ∈ (Pe*NI* ∪ P<sup>e</sup> *SP*,1) × e*<sup>λ</sup>*−, <sup>e</sup>*λ*<sup>+</sup> - \{0, 1} *. By using the quantities <sup>κ</sup><sup>λ</sup>* := *λκ*<sup>A</sup> + (1 − *<sup>λ</sup>*)*κ*<sup>H</sup> *and* <sup>Λ</sup>*<sup>λ</sup>* := q *λκ*<sup>2</sup> <sup>A</sup> + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*)*<sup>κ</sup>* 2 H *from* (150) *(which is well-defined, cf.* (138)*), one gets for all t* > 0

$$\mu(a) \qquad \lim\_{m \to \infty} m \cdot \left(1 - q\_{\lambda}^{(m)}\right) \\ = \lim\_{m \to \infty} m \cdot \left(1 - \beta\_{\lambda}^{(m)}\right) \\ = \frac{\kappa\_{\lambda}}{\sigma^2}.$$
 
$$\text{a. } (m) \qquad \text{a. } \quad \left(\begin{array}{c} (m) \\ \end{array}\right) \qquad (m) \qquad \lambda(1-\lambda)$$

(*b*) lim*m*→<sup>∞</sup> *m* 2 · *a* (*m*) <sup>1</sup> <sup>=</sup> lim*m*→<sup>∞</sup> *m* 2 · *q* (*m*) *<sup>λ</sup>* − *β* (*m*) *λ* = − *<sup>λ</sup>*(<sup>1</sup> − *<sup>λ</sup>*)(*κ*<sup>A</sup> − *<sup>κ</sup>*H) 2 2*σ* 4 = − Λ2 *<sup>λ</sup>* − *κ* 2 *λ* 2*σ* 4 . 

$$\mathbf{x}(\boldsymbol{\sigma}) = \lim\_{m \to \infty} m \cdot \mathbf{x}\_0^{(m)} = -\frac{\boldsymbol{\Lambda}\_{\boldsymbol{\Lambda}} - \boldsymbol{\kappa}\_{\boldsymbol{\Lambda}}}{\sigma^2} \left\{ \begin{array}{c} < \boldsymbol{0}, \quad \text{if} \quad \boldsymbol{\lambda} \in ]0, \mathbf{1}[\boldsymbol{\nu}] \\\\ > \boldsymbol{0}, \quad \text{if} \quad \boldsymbol{\lambda} \in ]\widetilde{\boldsymbol{\Lambda}}\_{-}, \widetilde{\boldsymbol{\lambda}}\_{+} \left[ \bigvee [0, 1] \right] \end{array} \right\}$$

$$(d) \qquad \lim\_{m \to \infty} m^2 \cdot \Gamma\_{<}^{(m)} = \lim\_{m \to \infty} m^2 \cdot \Gamma\_{>}^{(m)} = \frac{(\Lambda\_{\lambda} - \kappa\_{\lambda})^2}{2\sigma^4} > 0.1$$

$$\iota(e) = \lim\_{m \to \infty} m \cdot (1 - d^{(m), \mathcal{S}}) \ = \frac{\Lambda\_{\lambda} + \kappa\_{\lambda}}{2\sigma^2} > 0 \dots$$

$$(f) \qquad \lim\_{m \to \infty} m \cdot (1 - d^{(m),T}) = \frac{\Lambda\_{\lambda}}{\sigma^2} > 0 \,\,\,\lambda$$

$$\mathfrak{a}(\mathfrak{g}) = \lim\_{\substack{m \to \infty \\ \downarrow}} m \cdot (1 - d^{(m), \mathbf{S}} d^{(m), T}) = \frac{3\Lambda\_{\lambda} + \kappa\_{\lambda}}{2\sigma^{2}} > 0 \,\,\,\,\,\,$$

$$\ell(h) \qquad \lim\_{m \to \infty} \left( d^{(m),S} \right)^{\sigma^2 m t} = \exp \left\{ -\frac{\Lambda\_\lambda + \kappa\_\lambda}{2} \cdot t \right\} < 1 \ldots$$

$$\ell(i) \qquad \lim\_{m \to \infty} \left( d^{(m),T} \right)^{\sigma^2 m t} = \underset{\gamma \quad \dots}{\exp} \{-\Lambda\_{\Lambda} \cdot t\} < 1.$$

$$\ell(j) = \lim\_{\substack{m \to \infty \\ \ell(m) = 1}} \left( d^{(m),S} d^{(m),T} \right)^{\sigma^2 m t} = \exp \left\{ -\frac{3\Lambda\_\lambda + \kappa\_\lambda}{2} \cdot t \right\} < 1 \dots$$

(*k*) *for <sup>λ</sup>* ∈]0, 1[, *there holds for the respective quantities defined in* (142) *to* (145) lim*m*→<sup>∞</sup> *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*c = (Λ*<sup>λ</sup>* − *κλ*) 2 2*σ* <sup>2</sup> · <sup>Λ</sup>*<sup>λ</sup>* · *e* −Λ*λ*·*t* · 1 − *e* −Λ*λ*·*t* > 0 , lim*m*→<sup>∞</sup> *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c = 1 4 · Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> · 1 − *e* −Λ*λ*·*t* 2 > 0 , lim*m*→<sup>∞</sup> *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*<sup>c</sup> <sup>=</sup> (Λ*<sup>λ</sup>* − *κλ*) 2 *σ* 2 · " *e* − 1 2 (Λ*λ*+*κλ*)·*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λ*·*t* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* − *e* − 1 2 (Λ*λ*+*κλ*)·*t* 1 − *e* −Λ*λ*·*t* 2 · Λ*<sup>λ</sup>* # > 0 , lim*m*→<sup>∞</sup> *ϑ* (*m*) b*σ* <sup>2</sup>*mt*<sup>c</sup> <sup>=</sup> (Λ*<sup>λ</sup>* − *κλ*) 2 Λ*λ* · " 1 − *e* − 1 2 (3Λ*λ*+*κλ*)·*t* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* + *e* <sup>−</sup>Λ*λ*·*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* # > 0 .

$$\text{(1)}\qquad \text{for } \lambda \in \left[\widetilde{\lambda}\_{-}, \widetilde{\lambda}\_{+}\right] \text{[}\left[0, 1\right], \text{ there holds for the respective quantities defined in (146) to (149)}\qquad\text{(10.4)}$$

lim*m*→<sup>∞</sup> *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*c = (Λ*<sup>λ</sup>* − *κλ*) 2 2*σ* <sup>2</sup> · *<sup>κ</sup><sup>λ</sup>* · *e* −Λ*λ*·*t* · 1 − *e* −*κλ*·*t* > 0 , lim*m*→<sup>∞</sup> *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c = (Λ*<sup>λ</sup>* − *κλ*) 2 2 · *κ<sup>λ</sup>* · " 1 − *e* −Λ*λ*·*t* Λ*<sup>λ</sup>* − 1 − *e* −(Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* # > 0 , lim*m*→<sup>∞</sup> *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*<sup>c</sup> <sup>=</sup> (Λ*<sup>λ</sup>* − *κλ*) 2 2 · *σ* 2 · *e* − 1 2 (Λ*λ*+*κλ*)·*t* · " *t* − 1 − *e* −Λ*λ*·*t* Λ*<sup>λ</sup>* # > 0 , lim*m*→<sup>∞</sup> *ϑ* (*m*) b*σ* <sup>2</sup>*mt*<sup>c</sup> <sup>=</sup> (Λ*<sup>λ</sup>* <sup>−</sup> *<sup>κ</sup>λ*) 2 · " (Λ*<sup>λ</sup>* − *κλ*) 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* · (Λ*<sup>λ</sup>* + *κλ*) 2 + 1 − *e* − 1 2 (3Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* · (3Λ*<sup>λ</sup>* + *κλ*) − *e* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* · *t* # > 0 .

**Proof of Lemma A6**. For each of the assertions (a) to (l), we will make use of l'Hospital's rule. To begin with, we obtain for arbitrary *<sup>µ</sup>*, *<sup>ν</sup>* ∈ R

$$\begin{split} &\lim\_{m\to\infty}m\left[1-(\boldsymbol{\mathcal{P}}\_{\mathcal{A}}^{(m)})^{\mu}(\boldsymbol{\mathcal{P}}\_{\mathcal{H}}^{(m)})^{\nu}\right] \\ &=\lim\_{\boldsymbol{\mathcal{P}}}\frac{\boldsymbol{\kappa}\_{\mathcal{A}}}{\sigma^{2}}+\nu\frac{\boldsymbol{\kappa}\_{\mathcal{A}}}{\sigma^{2}}.\end{split}\tag{A71}$$

From this, the first part of (a) follows immediately and the second part is a direct consequence of the definition of *β* (*m*) *λ* . Part (b) can be deduced from (A71):

$$\begin{split} \lim\_{m \to \infty} m^2 \cdot a\_1^{(m)} &= \lim\_{m \to \infty} \frac{m}{2\sigma^2} \cdot \left[ \lambda \cdot \kappa\_{\mathcal{A}} \left( 1 - (\boldsymbol{\beta}\_{\mathcal{A}}^{(m)})^{\lambda - 1} (\boldsymbol{\beta}\_{\mathcal{H}}^{(m)})^{1 - \lambda} \right) \right. \\ &\left. + (1 - \lambda) \cdot \kappa\_{\mathcal{H}} \left( 1 - (\boldsymbol{\beta}\_{\mathcal{A}}^{(m)})^{\lambda} (\boldsymbol{\beta}\_{\mathcal{H}}^{(m)})^{-\lambda} \right) \right] = \ -\frac{\lambda (1 - \lambda) (\kappa\_{\mathcal{A}} - \kappa\_{\mathcal{H}})^2}{2\sigma^4} = -\frac{\Lambda\_{\lambda}^2 - \kappa\_{\lambda}^2}{2\sigma^4}. \end{split}$$

For the proof of (c), we rely on the inequalities *x* (*m*) <sup>0</sup> ≤ *x* (*m*) <sup>0</sup> ≤ *x* (*m*) 0 (*<sup>m</sup>* ∈ N), where *<sup>x</sup>* (*m*) 0 and *x* (*m*) 0 are the obvious notational adaptions of (124) and (126), respectively. Notice that *x* (*m*) 0 and *x* (*m*) 0 are solutions of the (again adapted) quadratic equations *Q* (*m*) *λ* (*x*) = *x* resp. *Q* (*m*) *λ* (*x*) = *x* (cf. (127) and (128)). These solutions clearly exist in the case *<sup>λ</sup>* ∈]0, 1[. For sufficiently large approximations steps *<sup>m</sup>* ∈ N, these solutions also exist in the case *<sup>λ</sup>* ∈ e*λ*−, e*λ*<sup>+</sup> -[0, 1] since (138) together with parts (a) and (b) imply

$$\lim\_{m \to \infty} \left( m \cdot (1 - q\_{\lambda}^{(m)}) \right)^2 - 2 \cdot q\_{\lambda}^{(m)} \cdot m^2 \cdot a\_1^{(m)} = \sigma^{-2} \cdot \left[ \lambda \mathbf{x}\_{\mathcal{A}}^2 + (1 - \lambda) \mathbf{x}\_{\mathcal{H}}^2 \right] \\ > 0, \qquad \text{for } \lambda \in \left[ \widetilde{\lambda}\_-, \widetilde{\lambda}\_+ \left[ \left\lfloor [0, 1] \right\rfloor \right] \right]$$

To prove part (c), we show that the limits of *x* (*m*) 0 and *x* (*m*) 0 coincide. Assume first that *<sup>λ</sup>* ∈]0, 1[. Using (a) and (b), we obtain together with the obvious limit lim*m*→<sup>∞</sup> *<sup>q</sup>* (*m*) *<sup>λ</sup>* = 1

$$\lim\_{m\to\infty} m \cdot \overline{\mathbf{x}}\_0^{(m)} = \lim\_{m\to\infty} \left( q\_\lambda^{(m)} \right)^{-1} \cdot \left[ m \cdot (1 - q\_\lambda^{(m)}) - \sqrt{\left( m \cdot (1 - q\_\lambda^{(m)}) \right)^2 - 2 \cdot q\_\lambda^{(m)} \cdot m^2 \cdot a\_1^{(m)}} \right]$$

$$= \frac{\kappa\_\lambda}{\sigma^2} - \sqrt{\left( \frac{\kappa\_\lambda}{\sigma^2} \right)^2 + \frac{\Lambda\_\lambda^2 - \kappa\_\lambda^2}{\sigma^4}} = -\frac{\Lambda\_\lambda - \kappa\_\lambda}{\sigma^2} \,. \tag{A72}$$

Let *x* (*m*) 0 be the adapted version of the auxiliary fixed-point lower bound defined in (125). By incorporating lim*m*→<sup>∞</sup> *<sup>β</sup>* (*m*) *<sup>λ</sup>* = <sup>1</sup> we obtain with (a) and (b)

$$\lim\_{m \to \infty} \underline{\boldsymbol{\omega}}\_{0}^{(m)} = \lim\_{m \to \infty} \max \left\{ -\boldsymbol{\beta}\_{\boldsymbol{\lambda}}^{(m)} \, , \, \frac{\boldsymbol{q}\_{\boldsymbol{\lambda}}^{(m)} - \boldsymbol{\beta}\_{\boldsymbol{\lambda}}^{(m)}}{1 - \boldsymbol{q}\_{\boldsymbol{\lambda}}^{(m)}} \right\} \\ = \lim\_{m \to \infty} \frac{1}{m} \cdot \frac{m^2 \cdot \boldsymbol{a}\_1^{(m)}}{m \cdot \left(1 - \boldsymbol{q}\_{\boldsymbol{\lambda}}^{(m)}\right)} \\ = 0,$$

which implies

$$\begin{split} \lim\_{m \to \infty} m \cdot \underline{\mathbf{x}}\_0^{(m)} &= \lim\_{m \to \infty} \frac{\underline{\sigma}^{-\frac{\mathbf{x}\_0^{(m)}}{\mathbf{0}}}}{q\_\lambda^{(m)}} \cdot \left[ m \cdot (1 - q\_\lambda^{(m)}) - \sqrt{\left( m \cdot (1 - q\_\lambda^{(m)}) \right)^2 - 2 \cdot \underline{\sigma}^{\underline{\mathbf{x}}^{(m)}} q\_\lambda^{(m)} \cdot m^2 \cdot \underline{a}\_1^{(m)}} \right] \\ &= \frac{\kappa\_\lambda}{\sigma^2} - \sqrt{\left( \frac{\kappa\_\lambda}{\sigma^2} \right)^2 + \frac{\Lambda\_\lambda^2 - \kappa\_\lambda^2}{\sigma^4}} = -\frac{\Lambda\_\lambda - \kappa\_\lambda}{\sigma^2} \,. \end{split} \tag{A73}$$

Combining (A72) and (A73), the desired result (c) follows for *<sup>λ</sup>* ∈]0, 1[. Assume now that *<sup>λ</sup>* ∈ e*λ*−, e*λ*<sup>+</sup> -[0, 1]. In this case the approximates *x* (*m*) 0 and *x* (*m*) 0 have a different form, given in (124) and (126). However, the calculations work out in the same way: with parts (a) and (b) we get

$$\begin{split} \lim\_{m \to \infty} m \cdot \mathbf{x}\_0^{(m)} &= \lim\_{m \to \infty} \frac{1}{q\_\lambda^{(m)}} \cdot \left[ m \cdot \left( 1 - q\_\lambda^{(m)} \right) - \sqrt{\left( m \cdot \left( 1 - q\_\lambda^{(m)} \right) \right)^2 - 2 \cdot q\_\lambda^{(m)} \cdot m^2 \cdot a\_1^{(m)}} \right] \\ &= \frac{\kappa\_\lambda}{\sigma^2} - \sqrt{\left( \frac{\kappa\_\lambda}{\sigma^2} \right)^2 + \frac{\Lambda\_\lambda^2 - \kappa\_\lambda^2}{\sigma^4}} = -\frac{\Lambda\_\lambda - \kappa\_\lambda}{\sigma^2} \end{split}$$

as well as

$$\begin{split} \lim\_{m \to \infty} m \cdot \mathbb{X}\_0^{(m)} &= \lim\_{m \to \infty} m \cdot \left( 1 - q\_{\lambda}^{(m)} \right) - \sqrt{\left( m \cdot \left( 1 - q\_{\lambda}^{(m)} \right) \right)^2 - 2 \cdot m^2 \cdot a\_1^{(m)}} \\ &= \frac{\kappa\_{\lambda}}{\sigma^2} - \sqrt{\left( \frac{\kappa\_{\lambda}}{\sigma^2} \right)^2 + \frac{\Lambda\_{\lambda}^2 - \kappa\_{\lambda}^2}{\sigma^4}} = -\frac{\Lambda\_{\lambda} - \kappa\_{\lambda}}{\sigma^2} \end{split}$$

which finally finishes the proof of part (c). Assertion (d) is a direct consequence of (c). Since the representations of the parameters *c* (*m*),*S* , *d* (*m*),*S* , *c* (*m*),*T* , *d* (*m*),*<sup>T</sup>* are the same in both cases *<sup>λ</sup>* <sup>∈</sup>]0, 1[ and *λ* ∈ e*λ*−, e*λ*<sup>+</sup> -[0, 1], the following considerations hold generally. Part (e) follows from (b) and (c) by

$$\lim\_{m \to \infty} m \cdot (1 - d^{(m), \mathcal{S}}) \ = \lim\_{m \to \infty} \frac{m^2 \cdot a\_1^{(m)}}{m \cdot \mathbf{x}\_0^{(m)}} \ = \frac{\Lambda\_\lambda + \kappa\_\lambda}{2\sigma^2} \ > \ \mathbf{0} \ .$$

Notice that this term is positive since on e*λ*−, e*λ*<sup>+</sup> - \{0, 1} there holds *<sup>κ</sup><sup>λ</sup>* > <sup>0</sup> as well as <sup>Λ</sup>*<sup>λ</sup>* > <sup>0</sup>, cf. (A70). To prove (f), we apply the general limit lim*x*→<sup>0</sup> *e <sup>x</sup>*−<sup>1</sup> *<sup>x</sup>* = 1 and get with (a), (c)

$$\lim\_{m \to \infty} m \cdot (1 - d^{(m),T}) = \lim\_{m \to \infty} \left( m \cdot \left( 1 - q\_{\lambda}^{(m)} \right) - q\_{\lambda}^{(m)} \cdot m \cdot x\_0^{(m)} \cdot \frac{\varepsilon^{x\_0^{(m)}} - 1}{x\_0^{(m)}} \right) \\ = \frac{\Lambda\_{\lambda}}{\sigma^2} \cdot x\_0^{(m)}$$

The limit (g) can be obtained from (e) and (f):

$$\lim\_{m \to \infty} m \cdot (1 - d^{(m), \mathcal{S}} d^{(m), T}) = \lim\_{m \to \infty} \left\{ m \cdot (1 - d^{(m), \mathcal{S}}) + d^{(m), \mathcal{S}} \cdot m \cdot (1 - d^{(m), T}) \right\} \\ = \frac{3\Lambda\_{\lambda} + \kappa\_{\lambda}}{2\sigma^{2}} \cdot \Lambda\_{\lambda}$$

The assertions (h) resp. (i) resp. (j) follow from (e) resp. (f) resp. (g) by using the general relation lim*m*→<sup>∞</sup> 1 + *xm m <sup>m</sup>* <sup>=</sup> exp {lim*m*→<sup>∞</sup> *<sup>x</sup>m*}. To get the last two parts (k) and (l), we make repeatedly use of the results (a) to (j) and combine them with the formulas (142) to (149) of Corollary 14. More detailed, for *<sup>λ</sup>* ∈]0, 1[ and thus *q* (*m*) *<sup>λ</sup>* < *β* (*m*) *λ* we obtain

$$\begin{array}{rcl} m \cdot \underline{\mathfrak{L}}\_{\lfloor \sigma^{2} m t \rfloor}^{(m)} & = & m^{2} \cdot \Gamma\_{<}^{(m)} \cdot \frac{\left( d^{(m),T} \right)^{\lfloor \sigma^{2} m t \rfloor} \cdot 1}{m \cdot \left( 1 - d^{(m),T} \right)} \cdot \left( 1 - \left( d^{(m),T} \right)^{\lfloor \sigma^{2} m t \rfloor} \right) \\ & \stackrel{m \to \infty}{\longrightarrow} \quad \frac{\left( \Lambda\_{\lambda} - \kappa\_{\lambda} \right)^{2}}{2 \sigma^{2} \cdot \Lambda\_{\lambda}} \cdot e^{-\Lambda\_{\lambda} \cdot t} \cdot \left( 1 - e^{-\Lambda\_{\lambda} \cdot t} \right) & > 0, \\\\ \underline{\mathfrak{L}}\_{\lfloor \sigma^{2} m t \rfloor}^{(m)} & = & m^{2} \cdot \Gamma\_{<}^{(m)} \cdot \frac{1 - \left( d^{(m),T} \right)^{\lfloor \sigma^{2} m t \rfloor}}{\left( m \cdot \left( 1 - d^{(m),T} \right) \right)^{2}} \cdot \left[ 1 - \frac{d^{(m),T} \left( 1 + \left( d^{(m),T} \right)^{\lfloor \sigma^{2} m t \rfloor} \right)}{1 + d^{(m),T}} \right] \\\\ & \stackrel{m \to \infty}{\longrightarrow} \quad \frac{1}{4} \cdot \left( \frac{\Lambda\_{\lambda} - \kappa\_{\lambda}}{\Lambda\_{\lambda}} \right)^{2} \cdot \left( 1 - e^{-\Lambda\_{\lambda} \cdot t} \right)^{2} > 0, \end{array}$$

$$\begin{array}{rcl} m \cdot \overline{\xi}^{(m)}\_{\left[\varepsilon^{2}mt\right]} &=& m^{2} \cdot \Gamma^{(m)}\_{<} \cdot \left[\frac{\left(d^{(m)}S\right)^{\left[\varepsilon^{2}mt\right]} - \left(d^{(m)}T\right)^{\left[\varepsilon^{2}mt\right]}}{m \cdot \left(1-d^{(m),T}\right) - m \cdot \left(1-d^{(m),S}\right)}\right.\\ &\left. \left. - \left(d^{(m),S}\right)^{\left[\varepsilon^{2}mt\right]-1} \cdot \frac{1-\left(d^{(m),T}\right)^{\left[\varepsilon^{2}mt\right]}}{m \cdot \left(1-d^{(m),T}\right)}\right] \\ &\xrightarrow{m \to \infty} \frac{\left(\Lambda\_{\lambda}-\kappa\_{\lambda}\right)^{2}}{\sigma^{2}} \cdot \left[\frac{\varepsilon^{-\frac{1}{2}(\Lambda\_{\lambda}+\kappa\_{\lambda})\cdot t}-e^{-\Lambda\_{\lambda}\cdot t}-\frac{e^{-\frac{1}{2}(\Lambda\_{\lambda}+\kappa\_{\lambda})\cdot t}\left(1-e^{-\Lambda\_{\lambda}\cdot t}\right)}{2\cdot\Lambda\_{\lambda}}\right] > 0,\end{array}$$

$$
\begin{split}
\overline{\theta}\_{\left[\sigma^{2}m^{\star}\right]}^{(m)} &= \quad \frac{m^{2} \cdot \Gamma\_{\leq}^{(m)} \cdot d^{(m),T}}{m \cdot \left(1 - d^{(m),T}\right)} \cdot \left[\frac{1 - \left(d^{(m),S}d^{(m),T}\right)^{\left[\sigma^{2}mt\right]}}{m \cdot \left(1 - d^{(m),S}d^{(m),T}\right)} - \frac{\left(d^{(m),S}\right)^{\left[\sigma^{2}mt\right]} - \left(d^{(m),T}\right)^{\left[\sigma^{2}mt\right]}}{m \cdot \left(1 - d^{(m),T}\right) - m \cdot \left(1 - d^{(m),S}\right)}\right] \\
& \stackrel{m \to \infty}{\longrightarrow} \quad \frac{\left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)^{2}}{\Lambda\_{\lambda}} \cdot \left[\frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}}{3\Lambda\_{\lambda} + \kappa\_{\lambda}} + \frac{e^{-\Lambda\_{\lambda} \cdot t} - e^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}}{\Lambda\_{\lambda} - \kappa\_{\lambda}}\right] &> 0 \,\,.
\end{split}
$$

For *<sup>λ</sup>* ∈ e*λ*−, e*λ*<sup>+</sup> -[0, 1] and thus *q* (*m*) *<sup>λ</sup>* > *β* (*m*) *λ* we get

*m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*c = *m* 2 · Γ (*m*) > · *d* (*m*),*T* b*<sup>σ</sup>* <sup>2</sup>*mt*c − *d* (*m*),*S* 2·b*<sup>σ</sup>* <sup>2</sup>*mt*c *m* · 1 − *d* (*m*),*S* <sup>1</sup> <sup>+</sup> *<sup>d</sup>* (*m*),*S* − *m* · 1 − *d* (*m*),*T <sup>m</sup>*→<sup>∞</sup> −→ (Λ*<sup>λ</sup>* − *κλ*) 2 2*σ* <sup>2</sup> · *<sup>κ</sup><sup>λ</sup>* · *e* −Λ*λ*·*t* · 1 − *e* −*κλ*·*t* > 0 , *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c = *m*2 · Γ (*m*) > *m* · 1 − *d* (*m*),*S* <sup>1</sup> <sup>+</sup> *<sup>d</sup>* (*m*),*S* − *m* · 1 − *d* (*m*),*T* · *d* (*m*),*T* · 1 − *d* (*m*),*T* b*<sup>σ</sup>* <sup>2</sup>*mt*c *m* · 1 − *d* (*m*),*T* − *d* (*m*),*S* 2 · 1 − *d* (*m*),*S* 2·b*<sup>σ</sup>* <sup>2</sup>*mt*c *m* · 1 − *d* (*m*),*S* <sup>1</sup> <sup>+</sup> *<sup>d</sup>* (*m*),*S <sup>m</sup>*→<sup>∞</sup> −→ (Λ*<sup>λ</sup>* − *κλ*) 2 2 · *κ<sup>λ</sup>* · " 1 − *e* −Λ*λ*·*t* Λ*<sup>λ</sup>* − 1 − *e* −(Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* # > 0 , *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*<sup>c</sup> <sup>=</sup> *<sup>m</sup>* 2 · Γ (*m*) > · *d* (*m*),*S* b*<sup>σ</sup>* <sup>2</sup>*mt*c−<sup>1</sup> · *σ* <sup>2</sup>*mt m* − 1 − *d* (*m*),*T* b*<sup>σ</sup>* <sup>2</sup>*mt*c *m* · 1 − *d* (*m*),*T <sup>m</sup>*→<sup>∞</sup> −→ (Λ*<sup>λ</sup>* − *κλ*) 2 2 · *σ* 2 · *e* − 1 2 (Λ*λ*+*κλ*)·*t* · " *t* − 1 − *e* −Λ*λ*·*t* Λ*<sup>λ</sup>* # > 0 ,

$$\begin{split} \overline{\sigma}\_{\left[\sigma^{2}mt\right]}^{(m)} &= \quad m^{2} \cdot \Gamma\_{>}^{(m)} \cdot \left[ \frac{m \cdot \left(1 - d^{(m),T}\right)}{m^{2} \cdot \left(1 - d^{(m),S}\right)^{2}} \cdot m \cdot \left(1 - d^{(m),S}\right)^{\left\lfloor\sigma^{2}mt\right\rfloor} \right) \\ &+ \frac{d^{(m),T} \left(1 - \left(d^{(m),S}d^{(m),T}\right)^{\left\lfloor\sigma^{2}mt\right\rfloor}\right)}{m \cdot \left(1 - d^{(m),T}\right) \cdot m \cdot \left(1 - d^{(m),S}d^{(m),T}\right)} - \frac{\left(d^{(m),S}\right)^{\left\lfloor\sigma^{2}mt\right\rfloor}}{m \cdot \left(1 - d^{(m),S}\right)} \cdot \frac{\left\lfloor\sigma^{2}mt\right\rfloor}{m} \Bigg] \\ &\overset{m \to \infty}{\longrightarrow} \qquad \left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)^{2} \cdot \left[\frac{\left(\Lambda\_{\lambda} - \kappa\_{\lambda}\right)\left(1 - \frac{\epsilon^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}}{\Lambda\_{\lambda} \cdot \left(3\Lambda\_{\lambda} + \kappa\_{\lambda}\right)} + \frac{1 - \frac{\epsilon^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda}) \cdot t}}{\Lambda\_{\lambda} + \kappa\_{\lambda}} \cdot t\right] > 0. \quad \square \end{split}$$

**Proof of Theorem 11.** It suffices to compute the limits of the bounds given in Corollary 14 as *m* tends to infinity. This is done by applying Lemma A6 which provides corresponding limits of all quantities of interest. Accordingly, for all *<sup>t</sup>* > 0 the lower bound (153) in the case *<sup>λ</sup>* ∈]0, 1[ can be obtained from (140), (142) and (143) by

lim*m*→<sup>∞</sup> exp ( *x* (*m*) 0 · " *X* (*m*) 0 − *η σ* 2 · *d* (*m*),*T* 1 − *d* (*m*),*T* # 1 − *d* (*m*),*T* b*<sup>σ</sup>* <sup>2</sup>*mt*c + *x* (*m*) 0 *η σ* 2 · j *σ* <sup>2</sup>*mt*<sup>k</sup> + *ζ* (*m*) b*σ* <sup>2</sup>*mt*c · *X* (*m*) <sup>0</sup> + *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c ) <sup>=</sup> lim*m*→<sup>∞</sup> exp ( *m* · *x* (*m*) 0 · *X* (*m*) 0 *m* − *η σ* 2 · *d* (*m*),*T m* · 1 − *d* (*m*),*T* 1 − *d* (*m*),*T* b*<sup>σ</sup>* <sup>2</sup>*mt*c + *m* · *x* (*m*) 0 *η σ* 2 · *σ* <sup>2</sup>*mt m* + *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*c · *X* (*m*) 0 *m* + *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c ) <sup>=</sup> exp ( − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 · *X*e<sup>0</sup> − *η σ* 2 · *σ* 2 Λ*λ* 1 − *e* −Λ*λt* − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 · *η σ* 2 · *σ* 2 *t* + (Λ*<sup>λ</sup>* − *κλ*) 2 2*σ* <sup>2</sup> · <sup>Λ</sup>*<sup>λ</sup>* · *e* −Λ*λ*·*t* · 1 − *e* −Λ*λ*·*t* · *X*e<sup>0</sup> + *η* 4*σ* 2 · Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> · 1 − *e* −Λ*λ*·*t* 2 ) <sup>=</sup> exp − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 *X*e<sup>0</sup> − *η* Λ*λ* 1 − *e* −Λ*λ*·*t* − *η σ* 2 (Λ*<sup>λ</sup>* − *κλ*) · *t* + *L* (1) *λ* (*t*) · *X*e<sup>0</sup> + *η σ* 2 · *L* (2) *λ* (*t*) .

For all *<sup>t</sup>* > <sup>0</sup>, the upper bound (154) in the case *<sup>λ</sup>* ∈]0, 1[ follows analogously from (141), (144), (145) by

lim*m*→<sup>∞</sup> exp ( *x* (*m*) 0 · " *X* (*m*) 0 − *η σ* 2 · *d* (*m*),*S* 1 − *d* (*m*),*S* # 1 − *d* (*m*),*S* b*<sup>σ</sup>* <sup>2</sup>*mt*c + *x* (*m*) 0 *η σ* 2 · j *σ* <sup>2</sup>*mt*<sup>k</sup> − *ζ* (*m*) b*σ* <sup>2</sup>*mt*c · *X* (*m*) <sup>0</sup> − *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c ) <sup>=</sup> lim*m*→<sup>∞</sup> exp ( *m* · *x* (*m*) 0 · *X* (*m*) 0 *m* − *η σ* 2 · *d* (*m*),*S m* · 1 − *d* (*m*),*S* 1 − *d* (*m*),*S* b*<sup>σ</sup>* <sup>2</sup>*mt*c + *m* · *x* (*m*) 0 *η σ* 2 · *σ* <sup>2</sup>*mt m* − *m* · *ζ* (*m*) b*σ* <sup>2</sup>*mt*c · *X* (*m*) 0 *m* − *ϑ* (*m*) b*σ* <sup>2</sup>*mt*c ) <sup>=</sup> exp ( − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 *X*e<sup>0</sup> − *η σ* 2 · 2*σ* 2 Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* 1 − *e* − 1 2 (Λ*λ*+*κλ*)*t* <sup>−</sup> Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 · *η σ* 2 · *σ* 2 *t* − (Λ*<sup>λ</sup>* − *κλ*) 2 *σ* 2 · " *e* − 1 2 (Λ*λ*+*κλ*)·*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λ*·*t* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* − *e* − 1 2 (Λ*λ*+*κλ*)·*t* 1 − *e* −Λ*λ*·*t* 2 · Λ*<sup>λ</sup>* # · *X*e<sup>0</sup> − *η σ* 2 (Λ*<sup>λ</sup>* − *κλ*) 2 Λ*λ* · " 1 − *e* − 1 2 (3Λ*λ*+*κλ*)·*t* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* + *e* <sup>−</sup>Λ*λ*·*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* # ) <sup>=</sup> exp ( − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 " *X*e<sup>0</sup> − *η* 1 2 (Λ*<sup>λ</sup>* + *κλ*) # 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* − *η σ* 2 (Λ*<sup>λ</sup>* − *κλ*) · *t* − *U* (1) *λ* (*t*) · *X*e<sup>0</sup> − *η σ* 2 · *U* (2) *λ* (*t*) ) .

In the case *<sup>λ</sup>* ∈ e*λ*−, e*λ*<sup>+</sup> -[0, 1], the lower bound as well as the upper bound of the Hellinger integral limit is obtained analogously, by taking into account that the quantities *ζ* (*m*) *n* , *ϑ* (*m*) *<sup>n</sup>* , *ζ* (*m*) *n* , *ϑ* (*m*) *<sup>n</sup>* now have the form (146) to (149) instead of (142) to (145). Thus, the functions *L* (1) *λ* (*t*), *U* (1) *λ* (*t*), *L* (2) *λ* (*t*), *U* (2) *λ* (*t*) are obtained by employing the limits of part (l) of Lemma A6 instead of part (k).

The next Lemma (and parts of its proof) will be useful for the verification of Theorem 12:

**Lemma A7.** *Recall the bounds on the Hellinger integral <sup>m</sup>*−*limit given in* (153) *and* (154) *of Theorem 11, in terms of L* (*i*) *λ* (*t*) *and U* (*i*) *λ* (*t*) *(i* = 1, 2*) defined by* (155) *to* (158)*. Correspondingly, one gets the following <sup>λ</sup>*−*limits for all <sup>t</sup>* ∈ [0, <sup>∞</sup>[*:*

*(a) for all <sup>κ</sup>*<sup>A</sup> ∈]0, <sup>∞</sup>[ *and all <sup>κ</sup>*<sup>H</sup> ∈ [0, <sup>∞</sup>[ *with <sup>κ</sup>*<sup>A</sup> 6= *<sup>κ</sup>*<sup>H</sup>

$$\lim\_{\lambda \nearrow 1} \frac{\partial L\_{\lambda}^{(1)}(t)}{\partial \lambda} = \lim\_{\lambda \nearrow 1} \frac{\partial L\_{\lambda}^{(2)}(t)}{\partial \lambda} = \lim\_{\lambda \nearrow 1} \frac{\partial L\_{\lambda}^{(1)}(t)}{\partial \lambda} = \lim\_{\lambda \nearrow 1} \frac{\partial L\_{\lambda}^{(2)}(t)}{\partial \lambda} = 0. \tag{A74}$$

*(b) for <sup>κ</sup>*<sup>A</sup> = <sup>0</sup> *and all <sup>κ</sup>*<sup>H</sup> ∈]0, <sup>∞</sup>[

$$\lim\_{\lambda \nearrow 1} \frac{\partial L\_{\lambda}^{(1)}(t)}{\partial \lambda} = -\frac{\kappa\_{\mathcal{H}}^2 \cdot t}{2\sigma^2} \,, \tag{A75}$$

$$\lim\_{\lambda \nearrow 1} \frac{\partial L\_{\lambda}^{(2)}(t)}{\partial \lambda} = -\frac{\kappa\_{\mathcal{H}}^2 \cdot t^2}{4} \,, \tag{A76}$$

$$\lim\_{\lambda \nearrow 1} \frac{\partial \, \mathcal{U}\_{\lambda}^{(1)}(t)}{\partial \lambda} = \lim\_{\lambda \nearrow 1} \frac{\partial \, \mathcal{U}\_{\lambda}^{(2)}(t)}{\partial \lambda} = \, \text{ } \text{ } \tag{A77}$$

**Proof of Lemma A7.** For all *<sup>κ</sup>*A, *<sup>κ</sup>*<sup>H</sup> <sup>∈</sup> [0, <sup>∞</sup>[ with *<sup>κ</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>κ</sup>*<sup>H</sup> one can deduce from (150) as well as (155) to (158) the following derivatives:

*∂L* (1) *λ* (*t*) *∂λ* <sup>=</sup> 1 2*σ* 2 ( *t* 2 Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> *κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 H h2*<sup>e</sup>* <sup>−</sup>2Λ*λ<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λt* i + *e* −Λ*λt* 1 − *e* −Λ*λt* Λ*λ* " Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 <sup>H</sup> <sup>−</sup> <sup>2</sup>Λ*λ*(*κ*<sup>A</sup> <sup>−</sup> *<sup>κ</sup>*H) − Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> *κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2 # ) , (A78) *∂L* (2) *λ* (*t*) *∂λ* <sup>=</sup> 1 4 ( Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* · 1 − *e* −Λ*λt* Λ*λ* !<sup>2</sup> · *κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 <sup>H</sup> <sup>−</sup> <sup>2</sup>Λ*λ*(*κ*<sup>A</sup> <sup>−</sup> *<sup>κ</sup>*H) <sup>−</sup> Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 H + *t* · *e* −Λ*λt* · Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> · 1 − *e* −Λ*λt* Λ*λ* · *κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 H ) , (A79) *∂ U* (1) (*t*) 1 ( *t* 

*λ ∂λ* <sup>=</sup> *σ* 2 Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* 2Λ*<sup>λ</sup> t e*−Λ*λ<sup>t</sup> κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 H − 2 *e* − 1 2 (Λ*λ*+*κλ*)*t κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 <sup>H</sup> <sup>+</sup> <sup>2</sup>Λ*λ*(*κ*<sup>A</sup> <sup>−</sup> *<sup>κ</sup>*H) − *e* − 1 2 (Λ*λ*+*κλ*)*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λt* 2Λ*<sup>λ</sup>* · *κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 <sup>H</sup> <sup>−</sup> <sup>2</sup>Λ*λ*(*κ*<sup>A</sup> <sup>−</sup> *<sup>κ</sup>*H) + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* 2Λ*<sup>λ</sup>* <sup>2</sup> " *t* 2 *e* − 1 2 (Λ*λ*+*κλ*)*t κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 <sup>H</sup> <sup>+</sup> <sup>2</sup>Λ*λ*(*κ*<sup>A</sup> <sup>−</sup> *<sup>κ</sup>*H) − *t* 2 *e* − 1 2 (3Λ*λ*+*κλ*)*t* 3 *κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 H + <sup>2</sup>Λ*λ*(*κ*<sup>A</sup> − *<sup>κ</sup>*H) +*e* − 1 2 (Λ*λ*+*κλ*)*t* · 1 − *e* −Λ*λt* Λ*λ* · *κ* 2 <sup>A</sup> − *<sup>κ</sup>* 2 H #

$$\left(1+\frac{\Lambda\_{\mathrm{h}}-\kappa\_{\mathrm{\lambda}}}{\Lambda\_{\mathrm{h}}}\left(\kappa\_{\mathrm{\mathcal{A}}}^{2}-\kappa\_{\mathrm{\mathcal{H}}}^{2}-2\Lambda\_{\mathrm{h}}(\kappa\_{\mathrm{\mathcal{A}}}-\kappa\_{\mathrm{\mathcal{H}}})\right)\left[\frac{e^{-\frac{1}{2}(\Lambda\_{\mathrm{h}}+\kappa\_{\mathrm{\lambda}})t}-e^{-\Lambda\_{\mathrm{\lambda}}t}}{\Lambda\_{\mathrm{h}}-\kappa\_{\mathrm{\lambda}}}-\frac{e^{-\frac{1}{2}(\Lambda\_{\mathrm{h}}+\kappa\_{\mathrm{\lambda}})t}\left(1-e^{-\Lambda\_{\mathrm{\lambda}}t}\right)}{2\Lambda\_{\mathrm{\lambda}}}\right]\right),\tag{A80}$$

*∂ U* (2) *λ* (*t*) *∂λ* <sup>=</sup> (Λ*<sup>λ</sup>* − *κλ*) 2 Λ*λ*(3Λ*<sup>λ</sup>* + *κλ*) " *t* 2 *e* − 1 2 (3Λ*λ*+*κλ*)*t* 3 *κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2Λ*<sup>λ</sup>* + *<sup>κ</sup>*<sup>A</sup> − *<sup>κ</sup>*<sup>H</sup> ! − 1 − *e* − 1 2 (3Λ*λ*+*κλ*)*t* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* · 3 *κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2Λ*<sup>λ</sup>* + *<sup>κ</sup>*<sup>A</sup> − *<sup>κ</sup>*<sup>H</sup> ! # + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* " *t* 2 *e* − 1 2 (Λ*λ*+*κλ*)*t κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2Λ*<sup>λ</sup>* + *<sup>κ</sup>*<sup>A</sup> − *<sup>κ</sup>*<sup>H</sup> ! <sup>−</sup> *t e*−Λ*λ<sup>t</sup> κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2Λ*<sup>λ</sup>* # + *e* − 1 2 (Λ*λ*+*κλ*)*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λt* Λ*λ κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2Λ*<sup>λ</sup>* − *<sup>κ</sup>*<sup>A</sup> + *<sup>κ</sup>*<sup>H</sup> ! + " 2 *κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2Λ*<sup>λ</sup>* − *<sup>κ</sup>*<sup>A</sup> + *<sup>κ</sup>*<sup>H</sup> ! − Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ<sup>2</sup> *λ* · *κ* 2 <sup>A</sup> <sup>−</sup> *<sup>κ</sup>* 2 H 2 # · 1 Λ*λ* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* 1 − *e* − 1 2 (3Λ*λ*+*κλ*)*t* − *e* − 1 2 (Λ*λ*+*κλ*)*<sup>t</sup>* + *e* −Λ*λt* . (A81)

If *<sup>κ</sup>*<sup>A</sup> <sup>∈</sup>]0, <sup>∞</sup>[ and *<sup>κ</sup>*<sup>H</sup> <sup>∈</sup> [0, <sup>∞</sup>[ with *<sup>κ</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>κ</sup>*H, then one gets lim*λ*%<sup>1</sup> <sup>Λ</sup>*<sup>λ</sup>* <sup>=</sup> lim*λ*%<sup>1</sup> *<sup>κ</sup><sup>λ</sup>* <sup>=</sup> *<sup>κ</sup>*<sup>A</sup> <sup>&</sup>gt; <sup>0</sup> which implies (A74) from (A78) to (A81). For the proof of part (b), let us correspondingly assume *<sup>κ</sup>*<sup>A</sup> <sup>=</sup> <sup>0</sup> and *<sup>κ</sup>*<sup>H</sup> <sup>∈</sup>]0, <sup>∞</sup>[, which by (150) leads to *<sup>κ</sup><sup>λ</sup>* <sup>=</sup> *<sup>κ</sup>*<sup>H</sup> · (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*), <sup>Λ</sup>*<sup>λ</sup>* <sup>=</sup> *<sup>κ</sup>*<sup>H</sup> · √ <sup>1</sup> − *<sup>λ</sup>* and the convergences lim*λ*%<sup>1</sup> <sup>Λ</sup>*<sup>λ</sup>* <sup>=</sup> lim*λ*%<sup>1</sup> *<sup>κ</sup><sup>λ</sup>* <sup>=</sup> <sup>0</sup>. From this, the assertions (A75), (A76), (A77) follow in a straightforward manner from (A78), (A79), (A80) – respectively – by using (parts of) the obvious relations

$$\lim\_{\lambda \nearrow 1} \frac{\kappa\_{\lambda}}{\Lambda\_{\lambda}} = 0, \qquad \lim\_{\lambda \nearrow 1} \frac{\Lambda\_{\lambda} \pm \kappa\_{\lambda}}{\Lambda\_{\lambda}} = \lim\_{\lambda \nearrow 1} \frac{\Lambda\_{\lambda} - \kappa\_{\lambda}}{\Lambda\_{\lambda} + \kappa\_{\lambda}} = 1,\tag{A82}$$

$$\lim\_{\lambda \nearrow 1} \frac{1 - e^{-\varepsilon\_{\lambda} \cdot t}}{\varepsilon\_{\lambda}} = \; t \qquad \text{for all } c\_{\lambda} \in \left\{ \Lambda\_{\lambda}, \frac{\Lambda\_{\lambda} + \kappa\_{\lambda}}{2}, \frac{3\Lambda\_{\lambda} + \kappa\_{\lambda}}{2} \right\}. \tag{A83}$$

In order to get the last assertion in (A77), we make use of the following limits

$$\lim\_{\lambda \nearrow 1} \frac{1}{\Lambda\_{\lambda} - \kappa\_{\lambda}} - \frac{3}{3\Lambda\_{\lambda} + \kappa\_{\lambda}} = \lim\_{\lambda \nearrow 1} \frac{4\kappa\chi}{(\kappa\_{\mathcal{H}} - \kappa\_{\mathcal{H}} \cdot \sqrt{1 - \lambda}) \cdot (3\kappa\_{\mathcal{H}} + \kappa\_{\mathcal{H}} \cdot \sqrt{1 - \lambda})} = \frac{4}{3\kappa\_{\mathcal{H}}}\tag{A84}$$

and

$$\lim\_{\lambda \nearrow 1} \frac{1}{\Lambda\_{\lambda}} \left[ \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\lambda} + \kappa\_{\lambda})t}}{3\Lambda\_{\lambda} + \kappa\_{\lambda}} - \frac{1 - e^{-\Lambda\_{\lambda}t}}{\Lambda\_{\lambda} - \kappa\_{\lambda}} + \frac{1 - e^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda})t}}{\Lambda\_{\lambda} - \kappa\_{\lambda}} \right] = 0. \tag{A85}$$

To see (A85), let us first observe that the involved limit can be rewritten as

$$\lim\_{\lambda \nearrow 1} \left\{ \frac{1}{\Lambda\_{\mathbb{A}}(\Lambda\_{\mathbb{A}} - \kappa\_{\lambda})} \left[ \frac{1}{3} - \frac{1}{3} \left. e^{-\frac{1}{2}(3\Lambda\_{\mathbb{A}} + \kappa\_{\lambda})t} + e^{-\Lambda\_{\mathbb{A}}t} - e^{-\frac{1}{2}(\Lambda\_{\mathbb{A}} + \kappa\_{\lambda})t} \right| \right. \tag{A86}$$

$$+\frac{1-e^{-\frac{1}{2}(3\Lambda\_{\mathrm{l}}+\kappa\_{\lambda})t}}{\Lambda\_{\mathrm{l}}}\left[\frac{1}{3\Lambda\_{\mathrm{l}}+\kappa\_{\lambda}}-\frac{1}{3(\Lambda\_{\mathrm{l}}-\kappa\_{\lambda})}\right]\Bigg\}.\tag{A87}$$

Substituting *x* := √ <sup>1</sup> − *<sup>λ</sup>* and applying l'Hospital's rule twice, we get for the first limit (A86)

lim *x*&0 1 <sup>3</sup> <sup>−</sup> <sup>1</sup> 3 *e* − *<sup>κ</sup>*H*<sup>t</sup>* 2 (3*x*+*x* 2 ) + *e* <sup>−</sup>*κ*H*tx* <sup>−</sup> *<sup>e</sup>* − *<sup>κ</sup>*H*<sup>t</sup>* 2 (*x*+*x* 2 ) *κ* 2 H · (*x* <sup>2</sup> − *<sup>x</sup>* 3) = lim *x*&0 *κ*H*t* 6 (3 + 2*x*) *e* − *<sup>κ</sup>*H*<sup>t</sup>* 2 (3*x*+*x* 2 ) <sup>−</sup> *<sup>κ</sup>*<sup>H</sup> *t e*−*κ*H*tx* <sup>+</sup> *κ*H*t* 2 (1 + 2*x*)*e* − *<sup>κ</sup>*H*<sup>t</sup>* 2 (*x*+*x* 2 ) *κ* 2 H · (2*x* − 3*x* 2) = lim *x*&0 h − *κ* 2 H*t* 2 <sup>12</sup> (3 + 2*x*) <sup>2</sup> + *κ*H*t* 3 i *e* − *<sup>κ</sup>*H*<sup>t</sup>* 2 (3*x*+*x* 2 ) + *κ* 2 H *t* 2 *e* <sup>−</sup>*κ*H*tx*<sup>−</sup> h *κ* 2 H*t* 2 4 (1 + 2*x*) <sup>2</sup> <sup>−</sup> *<sup>κ</sup>*<sup>H</sup> *<sup>t</sup>* i *e* − *<sup>κ</sup>*H*<sup>t</sup>* 2 (*x*+*x* 2 ) *κ* 2 H · (2 − 6*x*) = 1 2*κ* 2 H " − 3*κ* 2 H *t* 2 4 + *κ*H*t* 3 + *κ* 2 H*t* 2 − *κ* 2 H *t* 2 4 + *<sup>κ</sup>*H*<sup>t</sup>* # = 2*t* 3 *κ*<sup>H</sup> .

The second limit (A87) becomes

$$\lim\_{\lambda \nearrow 1} \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\mathbb{H}} + \kappa\_{\lambda})t}}{3\Lambda\_{\mathbb{H}} + \kappa\_{\lambda}} \cdot \frac{3\Lambda\_{\mathbb{A}} + \kappa\_{\lambda}}{\Lambda\_{\mathbb{H}}} \cdot \frac{-4\kappa\_{\mathcal{H}}}{(3\kappa\_{\mathcal{H}} + \sqrt{1 - \lambda}\kappa\_{\mathcal{H}})(3\kappa\_{\mathcal{H}} - 3\sqrt{1 - \lambda}\kappa\_{\mathcal{H}})} \tag{A88}$$

and consequently (A85) follows. To proceed with the proof of (A77), we rearrange

lim *λ*%1 *∂ U* (2) *λ* (*t*) *∂λ* <sup>=</sup> lim *λ*%1 ( Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> " Λ*λ* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup> t* 2 *e* − 1 2 (3Λ*λ*+*κλ*)*t* − 3*κ* 2 H 2Λ*<sup>λ</sup>* − *κ*<sup>H</sup> !! − Λ*λ* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* · 1 − *e* − 1 2 (3Λ*λ*+*κλ*)*t* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* − 3*κ* 2 H 2Λ*<sup>λ</sup>* − *κ*<sup>H</sup> ! + Λ*λ* Λ*<sup>λ</sup>* − *κ<sup>λ</sup> e* − 1 2 (Λ*λ*+*κλ*)*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λt* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* − *κ* 2 H 2Λ*<sup>λ</sup>* + *<sup>κ</sup>*<sup>H</sup> ! − Λ*λ* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* − *t* 2 *e* − 1 2 (Λ*λ*+*κλ*)*t* − *κ* 2 H 2Λ*<sup>λ</sup>* − *κ*<sup>H</sup> ! <sup>−</sup> *t e*−Λ*λ<sup>t</sup> κ* 2 H 2Λ*<sup>λ</sup>* ! # + " Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* −*κ* 2 <sup>H</sup> <sup>+</sup> <sup>2</sup>Λ*λκ*<sup>H</sup> + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> *κ* 2 H 2 # · " 1 − *e* − 1 2 (3Λ*λ*+*κλ*)*t* Λ*λ*(3Λ*<sup>λ</sup>* + *κλ*) − *e* − 1 2 (Λ*λ*+*κλ*)*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λt* Λ*λ*(Λ*<sup>λ</sup>* − *κλ*) # ) = lim *λ*%1 ( Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> " *κ* 2 H *t* 4 − 3 *e* − 1 2 (3Λ*λ*+*κλ*)*t* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* − *e* − 1 2 (Λ*λ*+*κλ*)*t* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* + 2 *e* −Λ*λt* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* ! (A89)

$$\begin{split} \boldsymbol{\lambda}\_{\boldsymbol{\lambda}} &+ \frac{\kappa\_{\mathcal{H}}^{2}}{2} \left( \frac{3\left(1 - e^{-\frac{1}{2}(3\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}} + \boldsymbol{\kappa}\_{\boldsymbol{\lambda}})t}\right)}{\left(3\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}} + \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)^{2}} - \frac{1 - e^{-\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}}t}}{\left(\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}} - \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)^{2}} + \frac{1 - e^{-\frac{1}{2}(\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}} + \boldsymbol{\kappa}\_{\boldsymbol{\lambda}})t}}{\left(\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}} - \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)^{2}} \right) \\ &\underbrace{\boldsymbol{\lambda}\_{\boldsymbol{\lambda}} \left(\boldsymbol{\kappa}\_{\boldsymbol{\lambda}} - \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)}\_{\left(\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}}, \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)} \underbrace{\boldsymbol{\lambda}\_{\boldsymbol{\lambda}} \left(\boldsymbol{\kappa}\_{\boldsymbol{\lambda}} - \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)}\_{\left(\boldsymbol{\Lambda}\_{\boldsymbol{\lambda}}, \boldsymbol{\kappa}\_{\boldsymbol{\lambda}}\right)} \end{split} \tag{A90}$$

+ *<sup>κ</sup>*<sup>H</sup> − Λ*λ* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* · *t e*<sup>−</sup> <sup>1</sup> 2 (3Λ*λ*+*κλ*)*t* 2 + Λ*λ* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* · 1 − *e* − 1 2 (3Λ*λ*+*κλ*)*t* 3Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* − Λ*λ* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* · *t e*<sup>−</sup> <sup>1</sup> 2 (Λ*λ*+*κλ*)*t* 2 + Λ*λ* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* · 1 − *e* −Λ*λt* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* − Λ*λ* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* · 1 − *e* − 1 2 (Λ*λ*+*κλ*)*t* Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* !# + " Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* −*κ* 2 <sup>H</sup> <sup>+</sup> <sup>2</sup>Λ*λκ*<sup>H</sup> + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* <sup>2</sup> *κ* 2 H 2 # · " 1 − *e* − 1 2 (3Λ*λ*+*κλ*)*t* Λ*λ*(3Λ*<sup>λ</sup>* + *κλ*) − *e* − 1 2 (Λ*λ*+*κλ*)*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>* −Λ*λt* Λ*λ*(Λ*<sup>λ</sup>* − *κλ*) # ). (A91)

By means of (A82) to (A84), the limit of the expression after the squared brackets in (A89) becomes

$$\lim\_{\lambda \nearrow 1} \left\{ \frac{\kappa\_{\mathcal{H}}^2 t}{4} \left[ \frac{1 - e^{-\frac{1}{2}(\Lambda\_{\mathbb{h}} + \kappa\_{\lambda})t}}{\Lambda\_{\mathbb{h}} - \kappa\_{\lambda}} - 2 \frac{1 - e^{-\Lambda\_{\mathbb{h}}t}}{\Lambda\_{\mathbb{h}} - \kappa\_{\lambda}} + 3 \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\mathbb{h}} + \kappa\_{\lambda})t}}{3\Lambda\_{\mathbb{h}} + \kappa\_{\lambda}} + \frac{1}{\Lambda\_{\mathbb{h}} - \kappa\_{\lambda}} - \frac{3}{3\Lambda\_{\mathbb{h}} + \kappa\_{\lambda}} \right] = \frac{\kappa\_{\mathcal{H}} t}{3}, \tag{A92}$$

and the limit of the expression in (A90) becomes with (A85)

$$\begin{split} \lim\_{\lambda \nearrow 1} \left\{ \frac{\Lambda\_{\lambda}}{\Lambda\_{\lambda} - \kappa\_{\lambda}} \cdot \frac{\kappa\_{\mathcal{H}}^{2}}{2\Lambda\_{\lambda}} \cdot \left[ \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\lambda} + \kappa\_{\lambda})t}}{3\Lambda\_{\lambda} + \kappa\_{\lambda}} - \frac{1 - e^{-\Lambda\_{\lambda}t}}{\Lambda\_{\lambda} - \kappa\_{\lambda}} + \frac{1 - e^{-\frac{1}{2}(\Lambda\_{\lambda} + \kappa\_{\lambda})t}}{\Lambda\_{\lambda} - \kappa\_{\lambda}} \right] \\ - \frac{\kappa\_{\mathcal{H}}^{2}}{2} \cdot \frac{1 - e^{-\frac{1}{2}(3\Lambda\_{\lambda} + \kappa\_{\lambda})t}}{3\Lambda\_{\lambda} + \kappa\_{\lambda}} \cdot \left[ \frac{1}{\Lambda\_{\lambda} - \kappa\_{\lambda}} - \frac{3}{3\Lambda\_{\lambda} + \kappa\_{\lambda}} \right] \end{split} \tag{A93}$$

By putting (A91)–(A93) together with (A85) we finally end up with

$$\lim\_{\lambda \nearrow 1} \frac{\partial \mathcal{U}\_{\lambda}^{(2)}(t)}{\partial \lambda} = \left[ \frac{\kappa\_{\mathcal{H}} t}{3} - \frac{\kappa\_{\mathcal{H}} t}{3} \right] + \kappa\_{\mathcal{H}} \left( -\frac{t}{6} + \frac{t}{6} - \frac{t}{2} + t - \frac{t}{2} \right) + \left[ -\kappa\_{\mathcal{H}}^2 + \frac{\kappa\_{\mathcal{H}}^2}{2} \right] \cdot 0 = 0, 1$$

which finishes the proof of Lemma A7.

**Proof of Theorem 12.** Recall from (131) the approximative Poisson offspring-distribution parameter *β* (*m*) • := 1 − *κ*• *σ* <sup>2</sup>*m* and Poisson immigration-distribution parameter *α* (*m*) • := *β* (*m*) • · *η σ* 2 , which is a special case of *β* (*m*) A , *β* (*m*) H , *α* (*m*) A , *α* (*m*) H ∈ PNI ∪ PSP,1. Let us first calculate lim*m*→<sup>∞</sup> *<sup>I</sup> P* (*m*) A,b*σ* <sup>2</sup>*mt*c *P* (*m*) H,b*σ* <sup>2</sup>*mt*c by starting

from Theorem 3(a). Correspondingly, we evaluate for all *<sup>κ</sup>*<sup>A</sup> <sup>≥</sup> <sup>0</sup>, *<sup>κ</sup>*<sup>H</sup> <sup>≥</sup> <sup>0</sup> with *<sup>κ</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>κ</sup>*<sup>H</sup> by a twofold application of l'Hospital's rule

$$\begin{split} &\lim\_{m\to\infty}m^{2}\cdot\left[\boldsymbol{\beta}\_{\mathcal{A}}^{(m)}\cdot\left(\log\left(\frac{\boldsymbol{\beta}\_{\mathcal{A}}^{(m)}}{\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}}\right)-1\right)+\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}\right] = \lim\_{m\to\infty}\frac{-m}{2\sigma^{2}}\left[\boldsymbol{\kappa}\_{\mathcal{A}}\log\left(\frac{\boldsymbol{\beta}\_{\mathcal{A}}^{(m)}}{\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}}\right)+\boldsymbol{\kappa}\_{\mathcal{H}}\left(1-\frac{\boldsymbol{\beta}\_{\mathcal{A}}^{(m)}}{\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}}\right)\right] \\ &= \ \frac{1}{2\sigma^{4}}\cdot\lim\_{m\to\infty}\frac{\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}\cdot\boldsymbol{\kappa}\_{\mathcal{A}}-\boldsymbol{\beta}\_{\mathcal{A}}^{(m)}\cdot\boldsymbol{\kappa}\_{\mathcal{H}}}{\left(\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}\right)^{2}}\cdot\left(\boldsymbol{\kappa}\_{\mathcal{A}}\cdot\frac{\boldsymbol{\beta}\_{\mathcal{H}}^{(m)}}{\boldsymbol{\beta}\_{\mathcal{A}}^{(m)}}-\boldsymbol{\kappa}\_{\mathcal{H}}\right) = \ \frac{\left(\boldsymbol{\kappa}\_{\mathcal{A}}-\boldsymbol{\kappa}\_{\mathcal{H}}\right)^{2}}{2\sigma^{4}}.\tag{A94} \end{split}$$

Additionally there holds

$$\lim\_{m \to \infty} m \cdot (1 - \beta\_{\mathcal{A}}^{(m)}) = \frac{\kappa\_{\mathcal{A}}}{\sigma^2} \quad \text{and} \quad \lim\_{m \to \infty} \left( \beta\_{\mathcal{A}}^{(m)} \right)^{\lfloor \sigma^2 m t \rfloor} = \lim\_{m \to \infty} \left[ \left( 1 - \frac{\kappa\_{\mathcal{A}}}{\sigma^2 m} \right)^m \right]^{\lfloor \sigma^2 m t \rfloor / m} = e^{-\kappa\_{\mathcal{A}} t}.\tag{A95}$$

For *<sup>κ</sup>*<sup>A</sup> <sup>&</sup>gt; <sup>0</sup>, we apply the upper part of formula (69) as well as (A94) and (A95) to derive

lim*m*→<sup>∞</sup> *Iλ P* (*m*) A,b*σ* <sup>2</sup>*mt*c *P* (*m*) H,b*σ* <sup>2</sup>*mt*c <sup>=</sup> lim*m*→<sup>∞</sup> *m*2 · *β* (*m*) A · log *β* (*m*) A *β* (*m*) H − 1 + *β* (*m*) H *m* · (1 − *β* (*m*) A ) · " *X* (*m*) 0 *m* − *α* (*m*) A *m* · (1 − *β* (*m*) A ) # · 1 − *β* (*m*) A b*<sup>σ</sup>* <sup>2</sup>*mt*c + *α* (*m*) A *β* (*m*) A · *m* · (1 − *β* (*m*) A ) · *m* 2 · " *β* (*m*) A · log *β* (*m*) A *β* (*m*) H ! − 1 ! + *β* (*m*) H # · *σ* <sup>2</sup>*mt m* # = (*κ*<sup>A</sup> − *κ*H) 2 2*σ* <sup>2</sup> · *<sup>κ</sup>*<sup>A</sup> · *X*e<sup>0</sup> <sup>−</sup> *η κ*A · 1 − *e* −*κ*A·*t* + *η* · *t* .

For *<sup>κ</sup>*<sup>A</sup> <sup>=</sup> <sup>0</sup> (and thus *<sup>κ</sup>*<sup>H</sup> <sup>&</sup>gt; <sup>0</sup>, *<sup>β</sup>* (*m*) <sup>A</sup> <sup>≡</sup> <sup>1</sup>, *<sup>α</sup>* (*m*) <sup>A</sup> <sup>≡</sup> *<sup>η</sup>*/*<sup>σ</sup>* 2 ), we apply the lower part of formula (69) as well as (A94) and (A95) to obtain

$$\lim\_{m\to\infty} I\_{\lambda} \left( \mathcal{P}\_{\mathcal{A}, \lfloor \mathcal{I}^{2mt} \rfloor}^{(m)} \Big| \Big| \Big| \mathcal{P}\_{\mathcal{H}, \lfloor \mathcal{I}^{2mt} \rfloor}^{(m)} \right) = \left\{ \lim\_{m\to\infty} m^2 \cdot \left[ \mathcal{P}\_{\mathcal{H}}^{(m)} - \log \mathcal{P}\_{\mathcal{H}}^{(m)} - 1 \right] \right.$$

$$\left. \cdot \left[ \frac{\eta}{2\sigma^2} \cdot \frac{\left( \lfloor \sigma^2 mt \rfloor \right)^2}{m^2} + \left( \frac{\mathcal{X}\_0^{(m)}}{m} + \frac{\eta}{2\sigma^2 \cdot m} \right) \cdot \frac{\left\lfloor \sigma^2 mt \right\rfloor}{m} \right] \right\} = \frac{\kappa\_{\mathcal{H}}^2}{2\sigma^2} \cdot \left[ \frac{\eta}{2} \cdot t^2 + \left. \widetilde{\mathcal{X}}\_0 \cdot t \right].$$

Let us now calculate the "converse" double limit

$$\lim\_{\lambda \nearrow 1} \lim\_{m \to \infty} I\_{\lambda} \left( P\_{\mathcal{A}\_{\mathbb{L}}[\mathcal{o}^{2}mt]}^{(m)} \middle| \middle| P\_{\mathcal{H}\_{\mathbb{L}}[\mathcal{o}^{2}mt]}^{(m)} \right) \\ = \lim\_{\lambda \nearrow 1} \lim\_{m \to \infty} \frac{1 - H\_{\lambda} \left( P\_{\mathcal{A}\_{\mathbb{L}}[\mathcal{o}^{2}mt]}^{(m)} \middle| \middle| P\_{\mathcal{H}\_{\mathbb{L}}[\mathcal{o}^{2}mt]}^{(m)} \right)}{\lambda \cdot (1 - \lambda)}.$$

This will be achieved by evaluating for each *t* > 0 the two limits

$$\lim\_{\lambda \nearrow 1} \frac{1 - D^L\_{\lambda, \tilde{X}\_0, t}}{\lambda \cdot (1 - \lambda)} \qquad \text{and} \qquad \lim\_{\lambda \nearrow 1} \frac{1 - D^L\_{\lambda, \tilde{X}\_0, t}}{\lambda \cdot (1 - \lambda)} \tag{A96}$$

which will turn out to coincide; the involved lower and upper bound *D<sup>L</sup> λ*,*X*e0,*t* , *D<sup>U</sup> λ*,*X*e0,*t* defined by (153) and (154) satisfy lim*λ*%<sup>1</sup> *<sup>D</sup><sup>L</sup> λ*,*X*e0,*t* <sup>=</sup> lim*λ*%<sup>1</sup> *<sup>D</sup><sup>U</sup> λ*,*X*e0,*t* = 1 as an easy consequence of the limits (cf. 150)

$$\lim\_{\lambda \nearrow 1} \Lambda\_{\lambda} = \mathbb{x}\_{\mathcal{A}} \ge 0 \qquad \text{and} \qquad \lim\_{\lambda \nearrow 1} \mathbb{x}\_{\lambda} = \mathbb{x}\_{\mathcal{A}} \ge 0 \,\,\,\tag{A97}$$

as well as the formulas (A82) and (A83) for the case *<sup>κ</sup>*<sup>A</sup> <sup>=</sup> <sup>0</sup>. Accordingly, we compute

lim *λ*%1 <sup>1</sup> <sup>−</sup> *<sup>D</sup><sup>L</sup> λ*,*X*e0,*t λ* · (1 − *λ*) = lim *λ*%1 <sup>−</sup>*D<sup>L</sup> λ*,*X*e0,*t* 1 − 2*λ ∂ ∂λ* " − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 · *X*e<sup>0</sup> − *η* Λ*λ* · 1 − *e* −Λ*λ*·*t* − *η σ* 2 · (Λ*<sup>λ</sup>* − *κλ*) · *t* + *L* (1) *λ* (*t*) · *X*e<sup>0</sup> + *η σ* 2 · *L* (2) *λ* (*t*) # = lim *λ*%1 ( − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 " *X*e<sup>0</sup> − *η* Λ*λ* · *te*−Λ*λ*·*<sup>t</sup>* · *∂* Λ*<sup>λ</sup> ∂λ* <sup>+</sup> 1 − *e* −Λ*λ*·*t* · *η* Λ<sup>2</sup> *λ* · *∂* Λ*<sup>λ</sup> ∂λ* # − 1 *σ* 2 · *∂ ∂λ* (Λ*<sup>λ</sup>* <sup>−</sup> *<sup>κ</sup>λ*) · *X*e<sup>0</sup> − *η* Λ*λ* · 1 − *e* −Λ*λ*·*t* − *η t σ* 2 · *∂ ∂λ* (Λ*<sup>λ</sup>* <sup>−</sup> *<sup>κ</sup>λ*) + *X*e<sup>0</sup> *∂L* (1) *λ* (*t*) *∂λ* <sup>+</sup> *η σ* 2 *∂L* (2) *λ* (*t*) *∂λ* ) , with (A98) 2 2

$$
\frac{\partial \Lambda\_{\lambda}}{\partial \lambda} = \frac{\kappa\_{\mathcal{A}}^2 - \kappa\_{\mathcal{H}}^2}{2\,\Lambda\_{\lambda}} \qquad \text{and} \qquad \frac{\partial \kappa\_{\lambda}}{\partial \lambda} = \kappa\_{\mathcal{A}} - \kappa\_{\mathcal{H}} \,. \tag{A99}
$$

For the case *<sup>κ</sup>*<sup>A</sup> <sup>&</sup>gt; <sup>0</sup>, one can combine this with (A97) and (A74) to end up with

$$\lim\_{\lambda \nearrow 1} \frac{1 - D^L\_{\lambda, \widetilde{\mathcal{X}}\_{0, t}}}{\lambda \cdot (1 - \lambda)} = \frac{\left(\varkappa\_{\mathcal{A}} - \varkappa\_{\mathcal{H}}\right)^2}{2\sigma^2 \cdot \varkappa\_{\mathcal{A}}} \cdot \left[ \left(\widetilde{\mathcal{X}}\_0 - \frac{\eta}{\varkappa\_{\mathcal{A}}}\right) \cdot \left(1 - e^{-\varkappa\_{\mathcal{A}} \cdot t}\right) + \eta \cdot t \right]. \tag{A100}$$

For the case *<sup>κ</sup>*<sup>A</sup> <sup>=</sup> <sup>0</sup>, we continue the calculation (A98) by rearranging terms and by employing the Formulas (A75), (A76), (A82) and (A83) as well as the obvious relation <sup>1</sup> Λ − Λ−*κ<sup>λ</sup>* <sup>Λ</sup><sup>2</sup> <sup>=</sup> <sup>1</sup> *κ*H and obtain

lim *λ*%1 <sup>1</sup> <sup>−</sup> *<sup>D</sup><sup>L</sup> λ*,*X*e0,*t λ* · (1 − *λ*) = lim *λ*%1 ( *κ* 2 H · *X*e<sup>0</sup> 2*σ* 2 " Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* · *t* · *e* <sup>−</sup>Λ*λ<sup>t</sup>* + 1 − *e* −Λ*λt* Λ*λ* # + *η* · *κ* 2 H · *t* 2*σ* 2 " 1 Λ*λ* − Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ<sup>2</sup> *λ* + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* · 1 − *e* −Λ*λt* Λ*λ* # − *η* · *κ* 2 H 2*σ* 2 · 1 − *e* −Λ*λt* Λ*λ* " 1 Λ*λ* − Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ<sup>2</sup> *λ* # − *κ*<sup>H</sup> · *X*e<sup>0</sup> *σ* 2 1 − *e* −Λ*λt* + *η* · *κ*<sup>H</sup> *σ* 2 " 1 − *e* −Λ*λt* Λ*λ* − *t* # + *∂L* (1) *λ* (*t*) *∂λ* · *<sup>X</sup>*e<sup>0</sup> <sup>+</sup> *η σ* 2 · *∂L* (2) *λ* (*t*) *∂λ* ) = *κ* 2 <sup>H</sup> *<sup>X</sup>*e<sup>0</sup> *<sup>t</sup> σ* 2 + *η κ*<sup>2</sup> H *t* 2*σ* 2 1 *κ*H + *t* − *η κ*<sup>H</sup> *t* 2*σ* 2 − *κ* 2 <sup>H</sup> *<sup>X</sup>*e<sup>0</sup> *<sup>t</sup>* 2*σ* 2 − *η κ*<sup>2</sup> H *t* 2 4*σ* 2 = *κ* 2 H 2*σ* 2 · h *η* 2 · *t* <sup>2</sup> <sup>+</sup> *<sup>X</sup>*e<sup>0</sup> · *<sup>t</sup>* i . (A101)

Let us now turn to the second limit (A96) for which we compute analogously to (A98)

lim *λ*%1 <sup>1</sup> <sup>−</sup> *<sup>D</sup><sup>U</sup> λ*,*X*e0,*t λ* · (1 − *λ*) = lim *λ*%1 <sup>−</sup>*D<sup>U</sup> λ*,*X*e0,*t* 1 − 2*λ ∂ ∂λ* " − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 · " *X*e<sup>0</sup> − *η* 1 2 (Λ*<sup>λ</sup>* + *κλ*) # · 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* − *η σ* 2 · (Λ*<sup>λ</sup>* − *κλ*) · *t* − *U* (1) *λ* (*t*) · *X*e<sup>0</sup> − *η σ* 2 · *U* (2) *λ* (*t*) # = lim *λ*%1 ( − Λ*<sup>λ</sup>* − *κ<sup>λ</sup> σ* 2 " *X*e<sup>0</sup> − *η* 1 2 (Λ*<sup>λ</sup>* + *κλ*) ! · *t* 2 · *e* − 1 2 (Λ*λ*+*κλ*)·*t ∂ ∂λ* (Λ*<sup>λ</sup>* <sup>+</sup> *<sup>κ</sup>λ*) + 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* · 2 · *η* (Λ*<sup>λ</sup>* + *κλ*) 2 · *∂ ∂λ* (Λ*<sup>λ</sup>* <sup>+</sup> *<sup>κ</sup>λ*) # − 1 *σ* 2 · *∂ ∂λ* (Λ*<sup>λ</sup>* <sup>−</sup> *<sup>κ</sup>λ*) · *X*e<sup>0</sup> − *η* 1 2 (Λ*<sup>λ</sup>* + *κλ*) ! · 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* − *η t σ* 2 · *∂ ∂λ* (Λ*<sup>λ</sup>* <sup>−</sup> *<sup>κ</sup>λ*) − *∂ U* (1) *λ* (*t*) *∂λ* · *<sup>X</sup>*e<sup>0</sup> <sup>−</sup> *η σ* 2 *∂ U* (2) *λ* (*t*) *∂λ* ) . (A102)

For the case *<sup>κ</sup>*<sup>A</sup> <sup>&</sup>gt; <sup>0</sup>, one can combine this with (A97), (A99) and (A74) to end up with

$$\lim\_{\lambda \nearrow 1} \frac{1 - D\_{\lambda, \widetilde{\mathbf{X}}\_{0,t}}^{\mathrm{II}}}{\lambda \cdot (1 - \lambda)} = \frac{\left(\mathbf{x}\_{\mathcal{A}} - \mathbf{x}\_{\mathcal{H}}\right)^2}{2\sigma^2 \cdot \mathbf{x}\_{\mathcal{A}}} \cdot \left[ \left(\widetilde{\mathbf{X}}\_0 - \frac{\eta}{\mathbf{x}\_{\mathcal{A}}}\right) \cdot \left(1 - e^{-\mathbf{x}\_{\mathcal{A}} \cdot t}\right) + \eta \cdot t \right]. \tag{A103}$$

For the case *<sup>κ</sup>*<sup>A</sup> <sup>=</sup> <sup>0</sup>, we continue the calculation of (A102) by rearranging terms and by employing the formulas (A77), (A82) and (A83) as well as the obvious relation lim*λ*%<sup>1</sup> 1 Λ*λ* − Λ*λ*−*κ<sup>λ</sup>* Λ*λ*(Λ*λ*+*κλ*) = <sup>2</sup> *κ*H to obtain

lim *λ*%1 <sup>1</sup> <sup>−</sup> *<sup>D</sup><sup>U</sup> λ*,*X*e0,*t λ* · (1 − *λ*) = lim *λ*%1 ( *t* · *X*e<sup>0</sup> 4*σ* 2 · Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ* · *e* − 1 2 (Λ*λ*+*κλ*)·*t κ* 2 <sup>H</sup> <sup>+</sup> <sup>2</sup>Λ*λκ*<sup>H</sup> + *X*e0 2*σ* 2 · 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*λ κ* 2 <sup>H</sup> <sup>−</sup> <sup>2</sup>Λ*λκ*<sup>H</sup> − *η* · *t σ* 2 *κ*H 1 + *e* − 1 2 (Λ*λ*+*κλ*)·*<sup>t</sup>* <sup>Λ</sup>*<sup>λ</sup>* − *<sup>κ</sup><sup>λ</sup>* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* − *κ* 2 H 2 · 1 Λ*λ* − Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ*(Λ*<sup>λ</sup>* + *κλ*) + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* · 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*λ* ! + 2*η σ* 2 · 1 − *e* − 1 2 (Λ*λ*+*κλ*)·*t* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* " *κ*H 1 + Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*<sup>λ</sup>* + *κ<sup>λ</sup>* − *κ* 2 H 2 1 Λ*λ* − Λ*<sup>λ</sup>* − *κ<sup>λ</sup>* Λ*λ*(Λ*<sup>λ</sup>* + *κλ*) # − *∂ U* (1) *λ* (*t*) *∂λ* · *<sup>X</sup>*e<sup>0</sup> <sup>−</sup> *η σ* 2 *∂ U* (2) *λ* (*t*) *∂λ* ) = *κ* 2 H *t X*e<sup>0</sup> 4*σ* 2 + *κ* 2 H *t X*e<sup>0</sup> 4*σ* 2 − *η t σ* 2 " 2*κ*<sup>H</sup> − *κ*<sup>H</sup> − *κ* 2 H *t* 4 # + *η t σ* 2 [2*κ*<sup>H</sup> − *<sup>κ</sup>*H] = *κ* 2 H 2*σ* 2 h *η* 2 · *t* <sup>2</sup> <sup>+</sup> *<sup>X</sup>*e<sup>0</sup> · *<sup>t</sup>* i . (A104)

Since (A100) coincides with (A103) and (A101) coincides with (A104), we have finished the proof.

#### **References**


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