*3.24. Upper Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1])

For the cases *λ* ∈ R\[0, 1], the investigation of upper bounds for the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*) is much easier than the above-mentioned derivations of lower bounds. In fact, we face a situation which is similar to the lower-bounds-studies for the cases *λ* ∈]0, 1[ : due to Properties 3 (P19), the function *φλ*(·) is strictly convex on the nonnegative real line. Furthermore, it is asymptotically linear, as stated in (P20). The monotonicity Properties 2 (P10) to (P12) imply that for the tightest upper bound (within our framework) one should use the parameters *p U λ* := *α λ* A *α* 1−*λ* <sup>H</sup> <sup>&</sup>gt; <sup>0</sup> and *q U λ* := *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; 0. Lemma A1 states that *<sup>p</sup> U <sup>λ</sup>* ≥ *α<sup>λ</sup>* resp. *q U <sup>λ</sup>* ≥ *βλ*, with equality iff *α*<sup>A</sup> = *α*<sup>H</sup> resp. iff *β*<sup>A</sup> = *β*H. From Properties 1 (P3a) we see that for *β*<sup>A</sup> 6= *β*<sup>H</sup> the corresponding sequence *a* (*q U λ* ) *n n*∈N is convergent to *x* (*q U λ* ) <sup>0</sup> ∈ ]0, − log(*q U λ* )] if *q U <sup>λ</sup>* ≤ min{1 , *e <sup>β</sup>λ*−1} (i.e., if *<sup>λ</sup>* <sup>∈</sup> [*λ*−, *<sup>λ</sup>*+], cf. Lemma 1 (a)), and otherwise it diverges to ∞ faster than exponentially (cf. (P3b)). If *β*<sup>A</sup> = *β*<sup>H</sup> (i.e., if (*β*A, *β*H, *α*A, *α*H) ∈ PSP,4 = PSP,4a ∪ PSP,4b), then one gets *q U <sup>λ</sup>* = *β<sup>λ</sup>* and *a* (*q U λ* ) *<sup>n</sup>* = 0 = *x* (*q U λ* ) 0 for all *n* ∈ N (cf. (P2)). Altogether, this leads to

**Proposition 14.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1]) *and all initial population sizes <sup>X</sup>*<sup>0</sup> <sup>∈</sup> <sup>N</sup> *there holds with p<sup>U</sup> λ* := *α λ* A *α* 1−*λ* H , *q U λ* := *β λ* A *β* 1−*λ* H

$$\mathcal{B}(\boldsymbol{a}) = \begin{array}{c} \mathcal{B}\_{\boldsymbol{\lambda}, \boldsymbol{X}\_{0}, 1}^{\mathrm{II}} = \widetilde{\mathcal{B}}\_{\boldsymbol{\lambda}, \boldsymbol{X}\_{0}, 1}^{(p\_{\boldsymbol{\lambda}}^{\mathrm{II}}, \boldsymbol{\beta}\_{\boldsymbol{\lambda}}^{\mathrm{II}})} = \exp\left\{ \left( \mathcal{B}\_{\boldsymbol{\lambda}}^{\boldsymbol{\lambda}} \mathcal{B}\_{\boldsymbol{\mathcal{H}}}^{1-\boldsymbol{\lambda}} - \mathcal{B}\_{\boldsymbol{\lambda}} \right) \cdot \mathcal{X}\_{0} + \mathcal{a}\_{\boldsymbol{\mathcal{A}}}^{\boldsymbol{\lambda}} \mathcal{a}\_{\boldsymbol{\mathcal{H}}}^{1-\boldsymbol{\lambda}} - \mathcal{a}\_{\boldsymbol{\lambda}} \right\} > 1, \\\ \boldsymbol{\lambda} = \boldsymbol{\lambda}\_{1} \end{array}$$

$$\begin{array}{ccccc}(b) & \text{the sequence } \left(\mathsf{B}\_{\lambda,\mathsf{X}\_{\mathsf{U}},\mathsf{n}}^{\mathsf{U}}\right)\_{\mathsf{n}\in\mathsf{N}} \text{ of upper bounds for } \mathsf{H}\_{\lambda}(\mathsf{P}\_{\mathcal{A},\mathsf{n}}||\mathsf{P}\_{\mathcal{H},\mathsf{n}}) \text{ given by} \\ & \begin{array}{c} \dots \ \ \end{array} \text{ for } \mathsf{V}\_{\mathsf{U}} \text{ and } \mathsf{V}\_{\mathsf{U}} \text{ are } \mathsf{V}\_{\mathsf{U}} \text{ and } \mathsf{V}\_{\mathsf{U}}. \\\end{array}$$

$$\boldsymbol{B}\_{\lambda,X\_{0},\boldsymbol{\eta}}^{\mathrm{II}} = \tilde{\boldsymbol{B}}\_{\lambda,X\_{0},\boldsymbol{\eta}}^{(p\_{\lambda}^{\mathrm{II}},q\_{\lambda}^{\mathrm{II}})} = \exp\left\{\boldsymbol{a}\_{\boldsymbol{\eta}}^{(q\_{\lambda}^{\mathrm{II}})} \cdot \mathbf{X}\_{0} + \sum\_{k=1}^{n} \boldsymbol{b}\_{k}^{(p\_{\lambda}^{\mathrm{II}},q\_{\lambda}^{\mathrm{II}})} \right\}$$

*is strictly increasing,*

$$\begin{array}{rcl} (\mathcal{C}) & \lim\_{\mathfrak{n}\to\infty} B^{\mathcal{U}}\_{\lambda, X\_0, \mathfrak{n}} & = & \infty \\\\ 1 & \dots & \quad \left\{ \begin{array}{rcl} \mathfrak{n}^{\mathcal{U}} \ . \textbf{exn} \end{array} \bigg\{ \mathfrak{n}^{(q^{\mathcal{U}}\_{\lambda})} \right\} \\_ . \end{array}$$

$$(d)\quad\lim\_{\mathfrak{n}\to\infty}\frac{1}{\mathfrak{n}}\log B\_{\lambda,X\_{0},\mathfrak{n}}^{\mathrm{U}} = \begin{cases} \begin{aligned} p\_{\lambda}^{\mathrm{II}} \cdot \exp\left\{ \mathbf{x}\_{0}^{(q\_{\lambda}^{\mathrm{II}})} \right\} - \mathfrak{a}\_{\lambda} &> 0, & \text{if } \lambda \in [\lambda\_{-}, \lambda\_{+}] \; \vert \; [0, 1] \; \vert \\\infty, & \text{if } \lambda \in [-\infty, \lambda\_{-}] \cup \vert \lambda\_{+}, \infty \vert \; \vert \end{aligned} \end{cases}$$

(*e*) *the map X*<sup>0</sup> 7→ *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* = *B*e (*p U λ* ,*q U λ* ) *λ*,*X*0,*n is strictly increasing*.

#### **4. Power Divergences of Non-Kullback-Leibler-Information-Divergence Type**

#### *4.1. A First Basic Result*

For orders *λ* ∈ R\{0, 1}, all the results of the previous Section 3 carry correspondingly over from the Hellinger integrals *Hλ*(·||·) to 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 the Renyi divergences *Rλ*(·||·), by virtue of the relation (cf. (7))

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

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

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

in the following, we concentrate on the latter. In particular, the above-mentioned carrying-over procedure leads to bounds on *I<sup>λ</sup>* (*P*A||*P*H) which are tighter than the general rudimentary bounds (cf. (10) and (11))

$$0 \le I\_{\lambda} \left( P\_{\mathcal{A}, \mathbb{H}} || P\_{\mathcal{H}, \mathbb{H}} \right) < \frac{1}{\lambda (1 - \lambda)}, \quad \text{for } \lambda \in ]0, 1[ \, \Big| \, \quad \quad 0 \le I\_{\lambda} \left( P\_{\mathcal{A}, \mathbb{H}} || P\_{\mathcal{H}, \mathbb{H}} \right) \le \infty, \quad \text{for } \lambda \in \mathbb{R} \left( [0, 1] \right).$$

Because power divergences have a *very insightful interpretation* as "directed distances" between two probability distributions (e.g., within our running-example epidemiological context), and function as important tools in statistics, information theory, machine learning, and artificial intelligence, we present explicitly the outcoming exact values respectively bounds of *I<sup>λ</sup>* (*P*A||*P*H) (*λ* ∈ R\{0, 1}, *n* ∈ N), in the current and the following subsections. For this, recall the case-dependent parameters *p <sup>A</sup>* = *p A <sup>λ</sup>* = *p <sup>A</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) and *<sup>q</sup> <sup>A</sup>* = *q A <sup>λ</sup>* = *q <sup>A</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) (*<sup>A</sup>* ∈ {*E*, *<sup>L</sup>*, *<sup>U</sup>*}). To begin with, we can deduce from Theorem 1

#### **Theorem 2.**

*(a) For all* (*β*A, *β*H, *α*A, *α*H) ∈ (P*NI* ∪ P*SP,1*)*, all initial population sizes X*<sup>0</sup> ∈ N0*, all observation horizons n* ∈ N *and all λ* ∈ R\{0, 1} *one can recursively compute the exact value*

$$I\_{\lambda}(P\_{\mathcal{A},\mathbb{II}}||P\_{\mathcal{H},\mathbb{II}}) = \frac{1}{\lambda(\lambda - 1)} \cdot \left[ \exp\left\{ a\_{\mathbb{II}}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \mathcal{X}\_{\mathbb{0}} + \frac{\mathfrak{a}\_{\mathcal{A}}}{\mathcal{B}\_{\mathcal{A}}} \sum\_{k=1}^{\mathfrak{n}} a\_{k}^{(q\_{\lambda}^{\mathbb{E}})} \right\} - 1 \right] =: V\_{\lambda, \mathcal{X}\_{0,\mathbb{II}}}^{\mathrm{I}} \,, \tag{65}$$

*where <sup>α</sup>*<sup>A</sup> *β*A *can be equivalently replaced by <sup>α</sup>*<sup>H</sup> *β*H *and q E λ* := *β λ* A *β* 1−*λ* H *. Notice that on* P*NI the formula* (65) *simplifies significantly, since α*<sup>A</sup> = *α*<sup>H</sup> = 0*.*

*(b) For general parameters p* ∈ R*, q* 6= 0 *recall the general expression (cf.* (42)*)*

$$\widetilde{\mathcal{B}}\_{\lambda, X\_0, \mu}^{(p, q)} := \exp \left\{ a\_n^{(q)} \cdot X\_0 + \frac{p}{q} \sum\_{k=1}^n a\_k^{(q)} + n \cdot \left( \frac{p}{q} \beta\_\lambda - a\_\lambda \right) \right\}$$

*as well as*

$$\tilde{B}^{(p,0)}\_{\lambda,X\_0,n} := \exp\left\{-\beta\_{\lambda} \cdot X\_0 + \left(p \cdot e^{-\beta\_{\lambda}} - \alpha\_{\lambda}\right) \cdot n\right\} \dots$$

*Then, for all* (*β*A, *β*H, *α*A, *α*H) ∈ P*SP*\P*SP,1, all λ* ∈ R\{0, 1}*, all coefficients p L λ* , *p U λ* , *q L λ* , *q U λ* ∈ R *which satisfy* (35) *for all x* ∈ N0*, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *one gets the following recursive bounds for the power divergences: for λ* ∈]0, 1[ *there holds*

$$I\_{\lambda}(\boldsymbol{P}\_{\mathcal{A},\boldsymbol{u}}||\boldsymbol{P}\_{\mathcal{H},\boldsymbol{u}}) \left\{ \begin{array}{c} < & \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \boldsymbol{B}^{L}\_{\lambda,X\_{0},\boldsymbol{u}}\right) \\\\ \geq & \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \boldsymbol{B}^{L}\_{\lambda,X\_{0},\boldsymbol{u}}\right) \\\\ \geq & \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \boldsymbol{B}^{L}\_{\lambda,X\_{0},\boldsymbol{u}}\right) \end{array} \right. \\\\ = & \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \min\left\{\widehat{\boldsymbol{B}}^{\left(p\_{\lambda}^{\boldsymbol{I}},\boldsymbol{d}\_{\lambda}^{\boldsymbol{I}}\right)}, 1\right\} \right) =: \mathcal{B}^{I,L}\_{\lambda,X\_{0},\boldsymbol{u}}. \end{array}$$

*whereas for λ* ∈ R\[0, 1] *there holds*

$$\begin{split} &I\_{\lambda}(\boldsymbol{P\_{\mathcal{A},\mathcal{U}}}||\boldsymbol{P\_{\mathcal{H},\mathcal{U}}}) \left\{ \begin{array}{c} < &\frac{1}{\lambda(\lambda-1)} \cdot \left(\boldsymbol{\mathcal{B}}\_{\lambda,\boldsymbol{X\_{0}\mathcal{U}}}^{\mathrm{II}} - 1\right) \\\\ \geq &\frac{1}{\lambda(\lambda-1)} \cdot \left(\boldsymbol{\mathcal{B}}\_{\lambda,\boldsymbol{X\_{0}\mathcal{U}}}^{\mathrm{I}} - 1\right) \end{array} =: \, \boldsymbol{\mathcal{B}}\_{\lambda,\boldsymbol{X\_{0}\mathcal{U}}}^{\mathrm{I},\mathrm{II}} \\\\ &\geq &\frac{1}{\lambda(\lambda-1)} \cdot \left(\boldsymbol{\mathcal{B}}\_{\lambda,\boldsymbol{X\_{0}\mathcal{U}}}^{\mathrm{I}} - 1\right) \ = \, \frac{1}{\lambda(\lambda-1)} \cdot \left(\max\left\{\widetilde{\boldsymbol{B}}\_{\lambda,\boldsymbol{X\_{0}\mathcal{U}}}^{\left(p\_{\lambda}^{\mathrm{I}},q\_{\lambda}^{\mathrm{I}}\right)} , 1\right\} - 1\right) =: \, \boldsymbol{\mathcal{B}}\_{\lambda,\boldsymbol{X\_{0}\mathcal{U}}}^{\mathrm{I},\mathrm{II}} \dots \, \frac{1}{\lambda(\lambda-1)} \end{split}$$

In order to deduce the subsequent *detailed* recursive analyses of power divergences, we also employ the obvious relations

$$\lim\_{n\to\infty}\frac{1}{n}\log\left(\frac{1}{\lambda(1-\lambda)} - I\_{\lambda}(P\_{\mathcal{A},\mathbb{H}}||P\_{\mathcal{H},\mathbb{H}})\right) \\ = \lim\_{n\to\infty}\frac{1}{n}\left[-\log\left(\lambda(1-\lambda)\right) + \log\left(H\_{\lambda}(P\_{\mathcal{A},\mathbb{H}}||P\_{\mathcal{H},\mathbb{H}})\right)\right]$$

$$= \lim\_{n\to\infty}\frac{1}{n}\log\left(H\_{\lambda}(P\_{\mathcal{A},\mathbb{H}}||P\_{\mathcal{H},\mathbb{H}})\right), \qquad \text{for }\lambda\in[0,1] \,,\tag{66}$$

as well as

$$\begin{split} \lim\_{n \to \infty} \frac{1}{n} \log \left( I\_{\lambda} (P\_{\mathcal{A}, \mathbb{z}} || P\_{\mathcal{H}, \mathbb{z}}) \right) &= \lim\_{n \to \infty} \frac{1}{n} \left[ -\log \left( \lambda (\lambda - 1) \right) + \log \left( H\_{\lambda} (P\_{\mathcal{A}, \mathbb{z}} || P\_{\mathcal{H}, \mathbb{z}}) - 1 \right) \right] \\ = \lim\_{n \to \infty} \frac{1}{n} \left[ \log \left( 1 - \frac{1}{H\_{\lambda} (P\_{\mathcal{A}, \mathbb{z}} || P\_{\mathcal{H}, \mathbb{z}})} \right) + \log \left( H\_{\lambda} (P\_{\mathcal{A}, \mathbb{z}} || P\_{\mathcal{H}, \mathbb{z}}) \right) \right] &= \lim\_{n \to \infty} \frac{1}{n} \log \left( H\_{\lambda} (P\_{\mathcal{A}, \mathbb{z}} || P\_{\mathcal{H}, \mathbb{z}}) \right), \end{split} \tag{67}$$

for *λ* ∈ R\[0, 1] (provided that lim inf*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) > 1).

*4.2. Detailed Analyses of the Exact Recursive Values of Iλ*(·||·)*, i.e., for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*NI* ∪ P*SP,1*) × (R\{0, 1})

**Corollary 2.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*NI*×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$I(a) = I\_{\lambda}(P\_{\mathcal{A},1}||P\_{\mathcal{H},1}) \\ = \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \exp\left\{ \left(\mathcal{J}\_{\mathcal{A}}^{\lambda}\mathcal{J}\_{\mathcal{H}}^{1-\lambda} - \beta\_{\lambda}\right) \cdot \mathbf{X}\_{0} \right\} \right) \\ > 0 \,, \quad$$

$$\begin{aligned}(b) \quad & \quad \text{the sequence } \left(I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n})\right)\_{n\in\mathbb{N}} \text{ given by} \\ & \quad I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \frac{1}{\lambda(1-\lambda)} \cdot \left(1-\exp\left\{a\_{n}^{(q\_{\lambda}^{\mathbb{F}})} \cdot X\_{0}\right\}\right) =: V\_{\lambda, X\_{0}n}^{I} \end{aligned}$$

*is strictly increasing,*

$$I(\mathcal{E}) = \lim\_{\substack{n \to \infty \\ \mathfrak{a} \to \infty}} I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \exp\left\{\mathbf{x}\_0^{(q\_{\lambda}^{\mathbb{E}})} \cdot \mathbf{X}\_0\right\}\right) \in \left[0, \frac{1}{\lambda(1-\lambda)}\right],$$

(*d*) lim*n*→<sup>∞</sup> 1 *n* log 1 *λ*(1 − *λ*) − *Iλ*(*P*A,*n*||*P*H,*n*) <sup>=</sup> lim*n*→<sup>∞</sup> 1 *n* log *Hλ*(*P*A,*n*||*P*H,*n*) = 0 ,

$$(e) \qquad \text{the map} \quad \mathcal{X}\_0 \mapsto \mathcal{V}^I\_{\lambda, \mathcal{X}\_0, \mathfrak{n}} \quad \text{is strictly increasing.}$$

**Corollary 3.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*NI* × (R\[0, 1]) *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$I(a) = I\_{\lambda}(P\_{\mathcal{A},1}||P\_{\mathcal{H},1}) \\
= \frac{1}{\lambda(\lambda - 1)} \cdot \left( \exp\left\{ \left( \beta\_{\mathcal{A}}^{\lambda} \beta\_{\mathcal{H}}^{1-\lambda} - \beta\_{\lambda} \right) \cdot X\_0 \right\} - 1 \right) \\
> 0 \cdot I\_{\lambda}$$

$$\begin{aligned}(b) \quad &\text{the sequence } \left(I\_{\lambda}(P\_{\mathcal{A},\mathbb{n}}||P\_{\mathcal{H},\mathbb{n}})\right)\_{\mathfrak{n}\in\mathbb{N}} \text{ given by} \\ &I\_{\lambda}(P\_{\mathcal{A},\mathbb{n}}||P\_{\mathcal{H},\mathbb{n}}) = \frac{1}{\lambda(\lambda-1)} \cdot \left(\exp\left\{a\_{\eta}^{(q\_{\lambda}^{\mathbb{E}})} \cdot X\_{0}\right\} - 1\right) \end{aligned}$$

*λ*(*λ* − 1) · *a <sup>n</sup>* · *X*<sup>0</sup> *is strictly increasing,*

$$(c) \qquad \lim\_{n \to \infty} I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},\mathcal{I}}) = \begin{cases} \frac{1}{\lambda(\lambda-1)} \cdot \left( \exp\left\{ \mathbf{x}\_{0}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \mathbf{X}\_{0} \right\} - 1 \right) > 0, & \text{if } \lambda \in [\lambda\_{-}, \lambda\_{+}] \backslash [0, 1], \\\infty, & \text{if } \lambda \in [-\infty, \lambda\_{-}] \cup [\lambda\_{+}, \infty], \end{cases}$$

$$(d)\qquad\lim\_{\mathfrak{n}\to\infty}\frac{1}{\mathfrak{n}}\log I\_{\lambda}(P\_{\mathcal{A},\mathfrak{n}}||P\_{\mathcal{H},\mathfrak{n}}) = \begin{cases} 0, & \text{if } \quad \lambda \in [\lambda\_{-}, \lambda\_{+}] \backslash [0, 1],\\\infty, & \text{if } \quad \lambda \in [-\infty, \lambda\_{-}] \cup [\lambda\_{+}, \infty] \end{cases}$$

$$(e) \qquad \text{the map} \quad \mathcal{X}\_0 \mapsto \begin{pmatrix} V\_{\lambda, X\_0, \mathfrak{n}} & \text{is strictly increasing.} \end{pmatrix}$$

**Corollary 4.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,1*×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$I(\boldsymbol{a}) = I\_{\lambda}(\boldsymbol{P\_{\mathcal{A},1}}||\boldsymbol{P\_{\mathcal{H},1}}) = \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \exp\left\{ \left(\boldsymbol{\beta}\_{\mathcal{A}}^{\lambda}\boldsymbol{\beta}\_{\mathcal{H}}^{1-\lambda} - \boldsymbol{\beta}\_{\lambda}\right) \cdot \left(\mathbf{X}\_{0} + \frac{\mathbf{a}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}}\right) \right\} \right) > 0\,\,\lambda$$

$$\begin{aligned}(b) \qquad &\text{the sequence } (I\_{\lambda}(\mathcal{P}\_{\mathcal{A},\mathbb{R}}||P\_{\mathcal{H},\mathbb{R}}))\_{\boldsymbol{n}\in\mathbb{N}} \text{ given by} \\ &I\_{\lambda}(\mathcal{P}\_{\mathcal{A},\mathbb{R}}||P\_{\mathcal{H},\mathbb{R}}) \ = \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \exp\left\{a\_{\boldsymbol{n}}^{(q\_{\lambda}^{\mathbb{R}})} \cdot \mathcal{X}\_{0} + \frac{a\_{\mathcal{A}}}{\beta\_{\mathcal{A}}} \sum\_{k=1}^{\mathcal{U}} a\_{k}^{(q\_{\lambda}^{\mathbb{R}})} \right\} \right) \ =: V\_{\lambda,\mathcal{X}\_{0},\mathbb{R}}^{I} \end{aligned}$$

=: *V I λ*,*X*0,*n*

*is strictly increasing,*

$$\mu(c) = \lim\_{n \to \infty} I\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n}) = \frac{1}{\lambda(1-\lambda)}\text{ .}$$

$$\mu(d) = \lim\_{\substack{n \to \infty \\ n}} \frac{1}{n} \log \left( \frac{1}{\lambda(1-\lambda)} - I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) \right) \\ = \frac{\mathfrak{a}\_{\mathcal{A}}}{\mathfrak{P}\_{\mathcal{A}}} \cdot \mathfrak{x}\_{0}^{(q\_{\lambda}^{\mathbb{E}})} < 0 \ \lambda$$

$$(e) \qquad \text{the map} \quad \mathcal{X}\_0 \mapsto \begin{pmatrix} V\_{\lambda, X\_0, \mathfrak{n}} & \text{is strictly increasing.} \end{pmatrix}$$

**Corollary 5.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,1* × (R\[0, 1]) *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$\begin{aligned} \mathbf{(a)} \quad & I\_{\lambda}(P\_{\mathcal{A},1}||P\_{\mathcal{H},1}) = \frac{1}{\lambda(\lambda-1)} \cdot \left( \exp\left\{ \left( \beta\_{\mathcal{A}}^{\lambda} \beta\_{\mathcal{H}}^{1-\lambda} - \beta\_{\lambda} \right) \cdot \left( \mathbf{X}\_{0} + \frac{a\_{\mathcal{A}}}{\beta\_{\mathcal{A}}} \right) \right\} - 1 \right) > 0, \\ \mathbf{(a)} \quad & \mathbf{(b)} \quad \mathbf{(c)} \quad \mathbf{(c)} \quad \mathbf{(d)} \quad \mathbf{(e)} \quad \mathbf{(e)} \quad \mathbf{(d)} \end{aligned}$$

$$\begin{aligned} (b) \qquad & \text{the sequence } (I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}))\_{n \in \mathbb{N}} \text{ given by} \\ & I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) \ = \frac{1}{\lambda(\lambda - 1)} \cdot \left( \exp\left\{ a\_{n}^{(q\_{\lambda}^{\mathbb{F}})} \cdot \mathbf{X}\_{0} + \frac{\mathbf{a}\_{\mathcal{A}}}{\beta\_{\mathcal{A}}} \sum\_{k=1}^{n} a\_{k}^{(q\_{\lambda}^{\mathbb{F}})} \right\} - 1 \right) \ =: V\_{\lambda, \mathbf{X}\_{0}, \mu}^{I} \end{aligned}$$

*is strictly increasing,*

$$(c) \qquad \lim\_{n \to \infty} I\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n}) \ = \infty,$$

$$(d)\qquad\lim\_{n\to\infty}\frac{1}{n}\log I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n})=\begin{cases}\frac{\mathfrak{a}\_{\mathcal{A}}}{\tilde{\mathcal{P}}\_{\mathcal{A}}}\cdot\mathfrak{a}\_{0}^{(q\_{\lambda}^{\mathbb{E}})} > 0, & \text{if}\quad\lambda\in[\lambda\_{-},\lambda\_{+}]\backslash[0,1],\\\infty, & \text{if}\quad\lambda\in[-\infty,\lambda\_{-}]\cup[\lambda\_{+},\infty].\end{cases}$$

$$\mathbf{x}^{\prime}(e) \qquad \text{the map} \quad \mathbf{X}\_{0} \mapsto \, \, V^{I}\_{\lambda\_{\prime}X\_{0}, \mathfrak{n}} \quad \text{is strictly increasing.}$$

In the assertions (a), (b), (d) of the Corollaries 4 and 5 the fraction *α*A/*β*<sup>A</sup> can be equivalently replaced by *α*H/*β*H.

Let us now derive the corresponding detailed results for the bounds of the power divergences for the parameter cases PSP\PSP,1, where the Hellinger integral, and thus *Iλ*(*P*A,*n*||*P*H,*n*), cannot be determined exactly. The extensive discussion on the Hellinger-integral bounds in the Sections 3.4–3.13, as well as in the Sections 3.16–3.24 can be carried over directly to obtain power-divergence bounds. In the following, we summarize the outcoming key results, referring a detailed discussion on the possible choices of *p A <sup>λ</sup>* = *p <sup>A</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) and *<sup>q</sup> A <sup>λ</sup>* = *q <sup>A</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) (*<sup>A</sup>* ∈ {*L*, *<sup>U</sup>*}) to the corresponding above-mentioned subsections.

*4.3. Lower Bounds of Iλ*(·||·) *for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[

**Corollary 6.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP,2* ∪ P*SP,3a* ∪ P*SP,3b*)×]0, 1[ *there exist parameters p U λ* , *q U λ which satisfy p U λ* ∈ - *α λ* A *α* 1−*λ* H , *α<sup>λ</sup> and q U λ* ∈ - *β λ* A *β* 1−*λ* H , *β<sup>λ</sup>* - *as well as* (35) *for all x* ∈ N0*, and for all such pairs* (*p U λ* , *q U λ* ) *and all initial population sizes X*<sup>0</sup> ∈ N *there holds*

$$\chi(a) = B\_{\lambda, X\_0, 1}^{I, L} = \frac{1}{\lambda (1 - \lambda)} \cdot \left(1 - \exp\left\{ \left(q\_{\lambda}^{\mathrm{II}} - \beta\_{\lambda}\right) \cdot X\_0 + \left. p\_{\lambda}^{\mathrm{II}} - a\_{\lambda} \right\} \right\} > 0,$$

(*b*) *the sequence B I*,*L λ*,*X*0,*n n*∈N *of lower bounds for Iλ*(*P*A,*n*||*P*H,*n*) *given by*

$$B\_{\\\lambda,\\
X\_{0,\\
u}}^{I,L} = \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \exp\left\{a\_n^{(q\_\lambda^{\mathrm{II}})} \cdot \mathcal{X}\_0 + \sum\_{k=1}^n b\_k^{(p\_\lambda^{\mathrm{II}}, q\_\lambda^{\mathrm{II}})} \right\} \right).$$

*is strictly increasing,*

(*c*) lim*n*→<sup>∞</sup> *B I*,*L <sup>λ</sup>*,*X*0,*<sup>n</sup>* <sup>=</sup> lim*n*→<sup>∞</sup> *<sup>I</sup>λ*(*P*A,*n*||*P*H,*n*) = <sup>1</sup> *λ*(1 − *λ*) ,

$$\mu(d) \qquad \lim\_{n \to \infty} \frac{1}{n} \log \left( \frac{1}{\lambda(1-\lambda)} - B^{I,L}\_{\lambda, X\_0, n} \right) \\ = p^{\mathcal{U}}\_{\lambda} \cdot e^{\mathbf{x}\_0^{(q^{\mathcal{U}}\_{\lambda})}} - \mathfrak{a}\_{\lambda} \\ < \mathbf{0}\_{\lambda}$$

$$(e)\qquad\qquad\text{the map}\quad\quad\mathbf{X}\_0 \mapsto \;\mathcal{B}^{I,L}\_{\lambda,\mathbf{X}\_0,\mathbf{n}}\quad\text{is strictly increasing.}$$

#### **Remark 4.**

*(a) Notice that in the case* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,2*× ]0, 1[*–where α λ* A *α* 1−*λ* <sup>H</sup> <sup>=</sup> *<sup>α</sup><sup>λ</sup>* <sup>=</sup> *<sup>α</sup>*<sup>A</sup> <sup>=</sup> *<sup>α</sup>*<sup>H</sup> <sup>=</sup> *<sup>α</sup>–we get the special choice p U <sup>λ</sup>* = *α and q U <sup>λ</sup>* = (*α* + *β*A) *λ* (*α* + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup> (cf. Section 3.7). For the constellations* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP,3a* ∪ P*SP,3b*)×]0, 1[ *there exist parameters*

$$\mathbb{P}\_{\lambda} \mathbb{P}\_{\lambda}^{\mathrm{II}} \in \left[ \mathbb{A}\_{\mathcal{A}}^{\lambda} \mathbb{A}\_{\mathcal{H}}^{1-\lambda}, \mathbb{A}\_{\lambda} \right], \eta\_{\lambda}^{\mathrm{II}} \in \left[ \mathbb{A}\_{\mathcal{A}}^{\lambda} \mathbb{A}\_{\mathcal{H}}^{1-\lambda}, \mathbb{A}\_{\lambda} \right] \text{ which satisfy (35) for all } \mathbf{x} \in \mathbb{N}\_{0}.$$


**Corollary 7.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4a*×]0, 1[ *there exist parameters p U <sup>λ</sup>* < *αλ,* 1 > *q U <sup>λ</sup>* > *β<sup>λ</sup>* = *β such that* (35) *is satisfied for all x* ∈ [0, ∞[ *and such that for all initial population sizes X*<sup>0</sup> ∈ N *at least the parts (c) and (d) of Corollary 6 hold true.*

As in Section 3.12, for the parameter setup (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,4b×]0, 1[ we cannot derive a lower bound for the power divergences which improves the generally valid lower bound *Iλ*(*P*A,*n*||*P*H,*n*) ≥ 0 (cf. (10)) by employing our proposed (*p U λ* , *q U λ* )-method.

*4.4. Upper Bounds of Iλ*(·||·) *for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[

Since in this setup the upper bounds of the power divergences can be derived from the lower bounds of the Hellinger integrals, we here appropriately adapt the results of Proposition 6.

**Corollary 8.** *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 *there holds with p<sup>L</sup> λ* := *α λ* A *α* 1−*λ* H *and q<sup>L</sup> λ* := *β λ* A *β* 1−*λ* H

$$\delta(a) = \mathcal{B}\_{\lambda, \mathbf{X}\_0, 1}^{III} = \frac{1}{\lambda(1 - \lambda)} \cdot \left(1 - \exp\left\{ \left(\mathcal{B}\_{\mathcal{A}}^{\lambda} \beta\_{\mathcal{H}}^{1 - \lambda} - \beta\_{\lambda}\right) \cdot \mathbf{X}\_0 + a\_{\mathcal{A}}^{\lambda} a\_{\mathcal{H}}^{1 - \lambda} - a\_{\lambda} \right\} \right) > 0,$$

(*b*) *the sequence of upper bounds B I*,*U λ*,*X*0,*n n*∈N *for Iλ*(*P*A,*n*||*P*H,*n*) *given by*

$$\begin{split} \mathcal{B}\_{\lambda,\mathbf{X}\_{0:\mathcal{U}}}^{IL} &= \frac{1}{\lambda(1-\lambda)} \cdot \left(1 - \exp\left\{a\_n^{(q\_{\lambda}^L)} \cdot \mathbf{X}\_0 + \frac{p\_\lambda^L}{q\_\lambda^L} \sum\_{k=1}^n a\_k^{(q\_{\lambda}^L)} + n \cdot \left(\frac{p\_\lambda^L}{q\_\lambda^L} \cdot \boldsymbol{\beta}\_\lambda - \boldsymbol{\alpha}\_\lambda\right)\right\}\right), \\ \text{is strictly increasing}, \end{split}$$

$$\mu(c) \quad \lim\_{n \to \infty} B\_{\lambda, X\_0, n}^{I, \mathrm{II}} \;= \; \frac{1}{\lambda(1 - \lambda)}\; \lambda$$

$$\mathbf{u}(d) \qquad \lim\_{n \to \infty} \frac{1}{n} \log \left( \frac{1}{\lambda(1-\lambda)} - \mathcal{B}\_{\lambda, \mathbf{X}\_0, \mathbf{u}}^{\mathrm{I}, \mathbf{U}} \right) \\ = \frac{p\_{\lambda}^{\mathrm{L}}}{q\_{\lambda}^{\mathrm{L}}} \cdot \left( \mathbf{x}\_0^{(q\_{\lambda}^{\mathrm{L}})} + \mathfrak{f}\_{\lambda} \right) - \mathfrak{a}\_{\lambda} \\ = p\_{\lambda}^{\mathrm{L}} \cdot \mathbf{e}^{\mathbf{x}\_0^{(q\_{\lambda}^{\mathrm{L}})}} - \mathfrak{a}\_{\lambda} < 0,$$

$$\text{In}(\mathcal{e}) \qquad \text{the map} \quad \mathcal{X}\_0 \longmapsto \mathcal{B}^{I,II}\_{\lambda, \mathcal{X}\_0, \mathfrak{n}} \quad \text{is strictly increasing.}$$

*4.5. Lower Bounds of Iλ*(·||·) *for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*)∈(P*SP*\P*SP,1*)×(R\[0,1])

In order to derive detailed results on lower bounds of the power divergences in the case *λ* ∈ R\[0, 1], we have to subsume and adapt the Hellinger-integral concerning lower-bounds investigations from the Sections 3.16–3.23. Recall the *λ*-sets ISP,2, ISP,3a, ISP,3b (cf. (58), (60), (62)). For the constellations PSP,2 × ISP,2 we employ the special choice *p L <sup>λ</sup>* = *α λ* A *α* 1−*λ* <sup>H</sup> <sup>=</sup> *<sup>α</sup><sup>λ</sup>* <sup>=</sup> *<sup>α</sup>*<sup>A</sup> <sup>=</sup> *<sup>α</sup>*<sup>H</sup> <sup>=</sup> *<sup>α</sup>* together with *q L <sup>λ</sup>* = (*α* + *β*A) *λ* (*α* + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup>* <sup>&</sup>gt; max{0, *<sup>β</sup>λ*} (cf. (58)) which satisfy (35) for all *x* ∈ N<sup>0</sup> and (56), whereas for the constellations (PSP,3a × ISP,3a)∪(PSP,3b × ISP,3b) we have proved the existence of parameters *p L λ* , *q L λ* satisfying both (35) for all *x* ∈ N<sup>0</sup> and (56) with two strict inequalities. Subsuming this, we obtain

**Corollary 9.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP,2* × I*SP,2*)∪(P*SP,3a* × I*SP,3a*)∪(P*SP,3b* × I*SP,3b*) *there exist parameters p L λ* , *q L <sup>λ</sup> which satisfy* max{0, *αλ*} ≤ *p L <sup>λ</sup>* ≤ *α λ* A *α* 1−*λ* H , max{0, *βλ*} < *q L <sup>λ</sup>* ≤ *β λ* A *β* 1−*λ* H *as well as* (35) *for all x* ∈ N0*, and for all such pairs* (*p L λ* , *q L λ* ) *and all initial population sizes X*<sup>0</sup> ∈ N *one gets*

$$\chi(a) = B\_{\lambda, X\_0, 1}^{I, L} = \frac{1}{\lambda \left(\lambda - 1\right)} \cdot \left(\exp\left\{ \left(q\_{\lambda}^L - \beta\_{\lambda}\right) \cdot X\_0 + p\_{\lambda}^L - a\_{\lambda} \right\} - 1\right) > 0\_{\lambda}$$

$$\text{If } (b) \qquad \text{the sequence } \left( \mathbf{B}\_{\lambda, X\_0, n}^{I, L} \right)\_{n \in \mathbb{N}} \text{ of lower bounds for } I\_{\lambda}(P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n}) \text{ given by}$$

$$B\_{\\\lambda,\\\\\lambda\_{0,\\\\\lambda},\\
\eta}^{I,L} = \frac{1}{\\\\\lambda(\\\\\lambda-1)} \cdot \left(\exp\left\{a\_{n}^{(q\_{\lambda}^{L})} \cdot \mathcal{X}\_{0} + \sum\_{k=1}^{n} b\_{k}^{(p\_{\lambda}^{L}q\_{\lambda}^{L})}\right\} - 1\right),$$

*is strictly increasing,*

$$\begin{aligned} (c) \qquad \lim\_{n \to \infty} B^{I,L}\_{\lambda, X\_0, n} &= \lim\_{n \to \infty} I\_{\lambda} (P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n}) &= \infty \text{ (} \lambda \text{)}\\ \qquad & \qquad \left\{ \begin{array}{ccccc} \end{array} \right. \quad \left\{ \begin{array}{c} (q\_1^{l}) \end{array} \right\} \end{array} $$

$$(d) \qquad \lim\_{\mathfrak{n}\to\infty} \frac{1}{\mathfrak{n}} \log B\_{\lambda, \mathbf{X}\_0, \mathfrak{n}}^{L, L} = \begin{cases} \ \operatorname{p}\_{\lambda}^{L} \cdot \exp\left\{ \operatorname{x}\_{0}^{(q\_{\lambda}^{L})} \right\} - a\_{\lambda} > 0, & \text{if} \quad q\_{\lambda}^{L} \le \min\left\{ 1; e^{\theta\_{\lambda} - 1} \right\}, \\\ \infty, & \text{if} \quad q\_{\lambda}^{L} > \min\left\{ 1; e^{\theta\_{\lambda} - 1} \right\}, \end{cases}$$

(*e*) *the map X*<sup>0</sup> 7→ *B I*,*L λ*,*X*0,*n is strictly increasing*.

Analogously to the discussions in the Sections 3.17–3.20, for the parameter setups PSP,2 × R\ ISP,2 ∪ [0, 1] ∪ PSP,3a × R\ ISP,3a ∪ [0, 1] ∪ PSP,3b × R\ ISP,3b ∪ [0, 1] ∪ PSP,3c × R\[0, 1] and for all initial population sizes *X*<sup>0</sup> ∈ N one can still show

$$0 < I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) \quad \quad \text{and} \quad \lim\_{n \to \infty} I\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \infty \dots$$

For the penultimate case we obtain

**Corollary 10.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4a* × (R\[0, 1]) *there exist parameters p L <sup>λ</sup>* > *α<sup>λ</sup> (where not necessarily p L <sup>λ</sup>* ≥ 0*) and* 0 < *q L <sup>λ</sup>* < *β<sup>λ</sup>* = *β*• < min{1,*e <sup>β</sup>*•−1} <sup>=</sup> *<sup>e</sup> β*•−1 *such that* (35) *is satisfied for all x* ∈ [0, ∞[ *and such that for all initial population sizes X*<sup>0</sup> ∈ N *at least the parts (c) and (d) of Corollary 9 hold true.*

Notice that for the last case (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,4b × R\[0, 1] (where (*β*<sup>A</sup> = *β*<sup>H</sup> ≥ 1) we cannot derive lower bounds of the power divergences which improve the generally valid lower bound *Iλ*(*P*A,*n*||*P*H,*n*) ≥ 0 (cf. (11)) by employing our proposed (*p U λ* , *q U λ* )-method.

*4.6. Upper Bounds of Iλ*(·||·) *for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1])

For these constellations we adapt Proposition 14, which after modulation becomes

**Corollary 11.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1]) *and all initial population sizes <sup>X</sup>*<sup>0</sup> <sup>∈</sup> <sup>N</sup> *there holds with p<sup>U</sup> λ* := *α λ* A *α* 1−*λ* H *and q<sup>U</sup> λ* := *β λ* A *β* 1−*λ* H

$$\delta(a) \qquad \mathcal{B}\_{\lambda, \mathbf{X}\_0, 1}^{I, II} = \frac{1}{\lambda(\lambda - 1)} \cdot \left( \exp \left\{ \left( \mathcal{B}\_{\mathcal{A}}^{\lambda} \mathcal{B}\_{\mathcal{H}}^{1 - \lambda} - \beta\_{\lambda} \right) \cdot \mathbf{X}\_0 + \mathcal{a}\_{\mathcal{A}}^{\lambda} \mathcal{a}\_{\mathcal{H}}^{1 - \lambda} - \mathfrak{a}\_{\lambda} \right\} - 1 \right) > 0,$$

(*b*) *the sequence B I*,*U λ*,*X*0,*n n*∈N *of upper bounds for Iλ*(*P*A,*n*||*P*H,*n*) *given by*

$$B\_{\\\lambda,\\\\X\_0,\\
n}^{I,II} = \frac{1}{\\\\\lambda(\\\\\lambda-1)} \cdot \left(\exp\left\{a\_n^{(q^{\mathrm{II}})} \cdot \mathcal{X}\_0 + \sum\_{k=1}^n b\_k^{(p^{\mathrm{II}}\_{\lambda} q\_{\lambda}^{\mathrm{II}})} \right\} - 1\right),$$

*is strictly increasing, B I*,*U*

$$\begin{aligned} (\mathbf{c}) \qquad & \lim\_{n \to \infty} B^{IJ}\_{\lambda, \mathbf{X}\_{0, \mathbf{I}}} = \infty \\ (d) \qquad & \lim\_{n \to \infty} \frac{1}{n} \log B^{IJ}\_{\lambda, \mathbf{X}\_{0, \mathbf{I}}} = \left\{ \begin{array}{c} p^{IJ}\_{\lambda} \cdot \exp \left\{ \mathbf{x}^{(q^{\mathbf{I}}\_{\lambda})}\_{0} \right\} - a\_{\lambda} > 0, & \text{if } \lambda \in [\lambda\_{-}, \lambda\_{+}] \backslash [0, 1], \\ \infty, & \text{if } \lambda \in [-\infty, \lambda\_{-}] \cup [\lambda\_{+}, \infty], \end{array} \right. \end{aligned}$$

(*e*) *the map X*<sup>0</sup> 7→ *B I*,*U λ*,*X*0,*n is strictly increasing*.

#### *4.7. Applications to Bayesian Decision Making*

As explained in Section 2.5, the power divergences fulfill

$$I\_{\lambda} \left( P\_{\mathcal{A}, \mathfrak{n}} || P\_{\mathfrak{H}, \mathfrak{n}} \right) = \int\_{0}^{1} \Delta \mathcal{B} \mathcal{R}\_{\widehat{\mathcal{L}\mathcal{D}}} \left( p\_{\mathcal{A}}^{\text{prior}} \right) \cdot \left( 1 - p\_{\mathcal{A}}^{\text{prior}} \right)^{\lambda - 2} \cdot \left( p\_{\mathcal{A}}^{\text{prior}} \right)^{-1 - \lambda} \text{d} p\_{\mathcal{A}}^{\text{prior}}, \qquad \lambda \in \mathbb{R}, \tag{cf. (21)},$$

and

$$I\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) = \lim\_{\substack{\chi \to p\_{\mathcal{A}}^{\text{prior}}}} \Delta \mathcal{B} \mathcal{R}\_{\mathcal{L} \mathcal{O}\_{\lambda,\chi}} \left( p\_{\mathcal{A}}^{\text{prior}} \right), \qquad \lambda \in ]0, 1[ \text{.} \tag{cf. (22)} \text{.}$$

and thus can be interpreted as (i) *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*, and as (ii) *limit* decision risk reduction (limit statistical information measure). Hence, by combining (21) and (22) with the investigations in the previous Sections 4.1–4.6, we obtain exact recursive values respectively recursive bounds of the above-mentioned decision risk reductions. For the sake of brevity, we omit the details here.

#### **5. Kullback-Leibler Information Divergence (Relative Entropy)**

### *5.1. Exact Values Respectively Upper Bounds of I*(·||·)

From (2), (3) and (6) in Section 2.4, one can immediately see that the Kullback-Leibler information divergence (relative entropy) between two competing Galton-Watson processes without/with immigration can be obtained by the limit

$$I(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \lim\_{\lambda \nearrow 1} I\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) \; , \tag{68}$$

and the reverse Kullback-Leibler information divergence (reverse relative entropy) by *I* (*P*H,*n*||*P*A,*n*) = lim*λ*&<sup>0</sup> *I<sup>λ</sup>* (*P*A,*n*||*P*H,*n*). Hence, in the following we concentrate only on (68), the reverse case works analogously. Accordingly, we can use (68) in appropriate combination with the *λ*∈]0, 1[-parts of the previous Section 4 (respectively, the corresponding parts of Section 3) in order to obtain detailed analyses for *I* (*P*H,*n*||*P*A,*n*). Let us start with the following assertions on exact values respectively upper bounds, which will be proved in Appendix A.2:

#### **Theorem 3.**

*(a) For all* (*β*A, *β*H, *α*A, *α*H) ∈ (P*NI* ∪ P*SP*,1)*, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *the Kullback-Leibler information divergence (relative entropy) is given by*

$$I(\mathcal{P}\_{\mathcal{A},\boldsymbol{n}}||\mathcal{P}\_{\mathcal{H},\boldsymbol{n}}) := I\_{\mathcal{X}\boldsymbol{\rho},\boldsymbol{n}} := \begin{cases} \frac{\boldsymbol{\beta}\_{\mathcal{A}'} \left(\log\left(\frac{\boldsymbol{\beta}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{H}}}\right) - 1\right) + \boldsymbol{\beta}\_{\mathcal{H}}}{1 - \boldsymbol{\beta}\_{\mathcal{A}}} \cdot \left[\boldsymbol{\mathcal{X}}\_{0} - \frac{\boldsymbol{a}\_{\mathcal{A}}}{1 - \boldsymbol{\beta}\_{\mathcal{A}}}\right] \cdot \left(1 - (\boldsymbol{\beta}\_{\mathcal{A}})^{\boldsymbol{n}}\right) \\\\ \quad + \frac{\boldsymbol{a}\_{\mathcal{A}'} \left[\boldsymbol{\beta}\_{\mathcal{A}'} \left(\log\left(\frac{\boldsymbol{\beta}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{H}}}\right) - 1\right) + \boldsymbol{\beta}\_{\mathcal{H}}\right]}{\boldsymbol{\beta}\_{\mathcal{A}} (1 - \boldsymbol{\beta}\_{\mathcal{A}})} \cdot \boldsymbol{n}, & \quad \text{if} \quad \boldsymbol{\beta}\_{\mathcal{A}} \neq 1, \end{cases} \tag{6.9}$$

*(b) For all* (*β*A, *β*H, *α*A, *α*H) ∈ P*SP*\P*SP,1, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N *there holds I*(*P*A,*n*||*P*H,*n*) ≤ *E U X*0,*n , where*

*E U X*0,*n* := *β*A· log *β*A *β*H −1 +*β*<sup>H</sup> <sup>1</sup>−*β*<sup>A</sup> · h *X*<sup>0</sup> − *α*A <sup>1</sup>−*β*<sup>A</sup> i · 1 − (*β*A) *n* + " *α*A· h *β*A· log *β*A *β*H −1 +*β*<sup>H</sup> i *<sup>β</sup>*A(1−*β*A) <sup>+</sup> *<sup>α</sup>*<sup>A</sup> h log *α*A*β*<sup>H</sup> *α*<sup>H</sup> *β*<sup>A</sup> − *β*H *β*A i + *α*<sup>H</sup> # · *n* , *if β*<sup>A</sup> 6= 1, [*β*<sup>H</sup> − log *β*<sup>H</sup> − 1] · - *α*A 2 · *n* <sup>2</sup> + *X*<sup>0</sup> + *α*A 2 · *n* + h *α*A h log *α*A*β*<sup>H</sup> *α*H − *β*<sup>H</sup> i + *α*<sup>H</sup> i · *n* , *if β*<sup>A</sup> = 1. (70)

#### **Remark 5.**

*(i) Notice that the exact values respectively upper bounds are in* closed form *(rather than in recursive form). (ii) The n*−*behaviour of (the bounds of) the Kullback-Leibler information divergence/relative entropy I*(*P*A,*n*||*P*H,*n*) *in Theorem 3 is influenced by the following facts:*


*with equality iff α*<sup>A</sup> = *α*<sup>H</sup> *and β*<sup>A</sup> = *β*H*.*

Again by using (68) in appropriate combination with the "*λ*∈]0, 1[-parts" of the previous Section 4 (respectively, the corresponding parts of Section 3), we obtain the following *(semi-)closed-form* lower bounds of *I* (*P*H,*n*||*P*A,*n*):

**Theorem 4.** *For all* (*β*A, *β*H, *α*A, *α*H) ∈ P*SP*\P*SP,1, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N

$$I(P\_{\mathcal{A},\mathbb{H}}||P\_{\mathcal{H},\mathbb{H}}) \ge \operatorname\*{\mathbb{E}}\_{\mathcal{X}\_{0,\mathbb{H}}}^{L} := \sup\_{k \in \mathbb{N}\_{0}, y \in [0,\infty]} \left\{ \operatorname\*{\mathbb{E}}\_{y,\mathcal{X}\_{0,\mathbb{H}}}^{\mathrm{L}\mathrm{an}}, \operatorname\*{\mathbb{E}}\_{k,\mathcal{X}\_{0},\mathbb{H}}^{\mathrm{L},\mathrm{sec}}, \operatorname{\mathbb{E}}\_{\mathcal{X}\_{0,\mathbb{H}}}^{\mathrm{L},\mathrm{hor}} \right\} \in \left[ \mathbf{0}, \infty[\text{.}\tag{71}$$

*where for all y* ∈ [0, ∞[ *we define the – possibly negatively valued– finite bound component*

*E L*,*tan y*,*X*0,*n* := h *<sup>β</sup>*<sup>A</sup> log *<sup>α</sup>*A+*β*A*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* + *β*<sup>H</sup> 1 − *<sup>α</sup>*A+*β*A*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* i · <sup>1</sup>−(*β*A) *n* <sup>1</sup>−*β*<sup>A</sup> · h *X*<sup>0</sup> − *α*A <sup>1</sup>−*β*<sup>A</sup> i + h *α*A *<sup>β</sup>*A(1−*β*A) h *<sup>β</sup>*<sup>A</sup> log *<sup>α</sup>*A+*β*A*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* + *β*<sup>H</sup> 1 − *<sup>α</sup>*A+*β*A*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* i + *α*<sup>H</sup> − *α*<sup>A</sup> *β*H *β*A <sup>1</sup> <sup>−</sup> *<sup>α</sup>*A+*β*A*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* i · *<sup>n</sup>* , *if <sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> 1, h log *<sup>α</sup>*A+*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* + *β*<sup>H</sup> 1 − *<sup>α</sup>*A+*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* i · - *α*A 2 · *n* <sup>2</sup> + *X*<sup>0</sup> + *α*A 2 · *n* + (*α*<sup>H</sup> − *α*A*β*H) 1 − *<sup>α</sup>*A+*<sup>y</sup> <sup>α</sup>*H+*β*H*<sup>y</sup>* · *n* , *if β*<sup>A</sup> = 1, (72)

*and for all k* ∈ N<sup>0</sup> *the – possibly negatively valued– finite bound component*

*E L*,*sec k*,*X*0,*n* := h *<sup>f</sup>*A(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)log *f*A(*k*+1) *f*H(*k*+1) <sup>−</sup> *<sup>f</sup>*A(*k*)log *f*A(*k*) *f*H(*k*) + *<sup>β</sup>*<sup>H</sup> − *<sup>β</sup>*<sup>A</sup> i · 1−(*β*A) *n* 1−*β*<sup>A</sup> · h *X*<sup>0</sup> − *α*A 1−*β*<sup>A</sup> i + h *α*A *β*A(1−*β*A) *<sup>f</sup>*A(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)log *f*A(*k*+1) *f*H(*k*+1) <sup>−</sup> *<sup>f</sup>*A(*k*)log *f*A(*k*) *f*H(*k*) + *<sup>β</sup>*<sup>H</sup> − *<sup>β</sup>*<sup>A</sup> − *<sup>f</sup>*A(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)log *f*A(*k*+1) *f*H(*k*+1) <sup>−</sup> *<sup>f</sup>*A(*k*)log *f*A(*k*) *f*H(*k*) · *k* + *α*A *β*A <sup>+</sup>*f*A(*k*)log *f*A(*k*) *f*H(*k*) − *α*A*β*<sup>H</sup> *β*A + *<sup>α</sup>*<sup>H</sup> i · *<sup>n</sup>* , *if <sup>β</sup>*<sup>A</sup> 6= 1, h *<sup>f</sup>*A(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)log *f*A(*k*+1) *f*H(*k*+1) <sup>−</sup> *<sup>f</sup>*A(*k*)log *f*A(*k*) *f*H(*k*) + *<sup>β</sup>*<sup>H</sup> − <sup>1</sup> i · - *α*A 2 · *n* <sup>2</sup> + *X*<sup>0</sup> + *α*A 2 · *n* − h *<sup>f</sup>*A(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)log *f*A(*k*+1) *f*H(*k*+1) <sup>−</sup> *<sup>f</sup>*A(*k*)log *f*A(*k*) *f*H(*k*) (*<sup>k</sup>* <sup>+</sup> *<sup>α</sup>*A) <sup>−</sup>*f*A(*k*)log *f*A(*k*) *f*H(*k*) + *<sup>α</sup>*A*β*<sup>H</sup> − *<sup>α</sup>*<sup>H</sup> i · *<sup>n</sup>* , *if <sup>β</sup>*<sup>A</sup> = 1. (73)

*Furthermore, on* <sup>P</sup>*SP,4 we set EL*,*hor X*0,*n* := 0 *for all n* ∈ N *whereas on* P*SP*\(P*SP,1* ∪ P*SP,4*) *we define*

$$E\_{X\_0, \mathbb{H}}^{L,hor} := \left[ (\mathfrak{a}\_{\mathcal{A}} + \mathfrak{f}\_{\mathcal{A}} z^\*) \cdot \left[ \log \left( \frac{\mathfrak{a}\_{\mathcal{A}} + \mathfrak{f}\_{\mathcal{A}} z^\*}{\mathfrak{a}\_{\mathcal{H}} + \mathfrak{f}\_{\mathcal{H}} z^\*} \right) - 1 \right] + \mathfrak{a}\_{\mathcal{H}} + \mathfrak{f}\_{\mathcal{H}} z^\* \right] \cdot \mathfrak{n}, \qquad , \mathfrak{n} \in \mathbb{N}, \tag{74}$$

*with z*∗ := arg max*x*∈N<sup>0</sup> n (*α*<sup>A</sup> + *β*A*x*) h <sup>−</sup> log *<sup>α</sup>*A+*β*A*<sup>x</sup> <sup>α</sup>*H+*β*H*<sup>x</sup>* + 1 i − (*α*<sup>H</sup> + *β*H*x*) o *. On* <sup>P</sup>*SP*\(P*SP,1* ∪ P*SP,3c*) *one even gets E<sup>L</sup> <sup>X</sup>*0,*<sup>n</sup>* > 0 *for all X*<sup>0</sup> ∈ N *and all n* ∈ N*. For the subcase* P*SP,3c, one obtains for each fixed n* ∈ N *and each fixed X*<sup>0</sup> ∈ N *the strict positivity E L <sup>X</sup>*0,*<sup>n</sup>* > 0 *if ∂ ∂y E L*,*tan y*,*n* (*y* ∗ ) 6= 0*, where y*<sup>∗</sup> := *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> ∈ N *and hence*

$$= \begin{pmatrix} \frac{\partial}{\partial y} E\_{y,\mathcal{A}\mu}^{\perp,\text{far}} \\ \frac{\partial}{\partial \boldsymbol{\mu}} y\_{\mathcal{A}\mu} \boldsymbol{n} \end{pmatrix} (\boldsymbol{y}^\*) \tag{75}$$
 
$$= \begin{cases} -\frac{(\beta\_A - \beta\_{\mathcal{H}})^3}{\overline{a}\_A \beta\_{\mathcal{H}} - a\_{\mathcal{H}} \beta\_A} \cdot \frac{1 - (\beta\_A)^3}{1 - \beta\_A} \cdot \left[ \mathbf{X}\_0 - \frac{\mathbf{a}\_A}{1 - \beta\_A} \right] - \frac{(\beta\_A - \beta\_{\mathcal{H}})^2}{\beta\_A} \left( 1 + \frac{\overline{a}\_A (\beta\_A - \beta\_{\mathcal{H}})}{(1 - \beta\_A)(a\_A \beta\_{\mathcal{H}} - a\_{\mathcal{H}} \beta\_A)} \right) \cdot \boldsymbol{n}\_\prime & \text{if } \beta\_{\mathcal{A}} \neq \mathbf{1}\_\prime \\\ -\frac{(1 - \beta\_{\mathcal{H}})^3}{\overline{a}\_A \beta\_{\mathcal{H}} - a\_{\mathcal{H}}} \cdot \left[ \frac{\underline{a}\_A}{2} \cdot \boldsymbol{n}^2 + \left( \mathbf{X}\_0 + \frac{\mathbf{a}\_A}{2} \right) \cdot \boldsymbol{n} \right] - (1 - \beta\_{\mathcal{H}})^2 \cdot \boldsymbol{n}\_\prime & \text{if } \beta\_{\mathcal{A}} = \mathbf{1}\_\prime \end{cases}$$

A proof of this theorem is given in in Appendix A.2.

**Remark 6.** *Consider the exemplary parameter setup* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) = ( <sup>1</sup> 3 , 2 3 , 2, 1) ∈ P*SP,3c; within our running-example epidemiological context of Section 2.3, this corresponds to a "semi-mild" infectious-disease-transmission situation* (H) *(with subcritical reproduction number <sup>β</sup>*<sup>H</sup> <sup>=</sup> <sup>2</sup> 3 *and importation mean of <sup>α</sup>*<sup>H</sup> <sup>=</sup> <sup>1</sup>*), whereas* (A) *describes a "mild" situation (with "low" subcritical <sup>β</sup>*<sup>A</sup> <sup>=</sup> <sup>1</sup> 3 *and α*<sup>A</sup> = 2*). In the case of <sup>X</sup>*<sup>0</sup> <sup>=</sup> <sup>3</sup> *there holds ∂ ∂y E L*,*tan y*,*X*0,*n* (*y* ∗ ) = 0 *for all n* ∈ N*, whereas for X*<sup>0</sup> 6= 3 *one obtains ∂ ∂y E L*,*tan y*,*X*0,*n* (*y* ∗ ) 6= 0 *for all n* ∈ N*.*

It seems that the optimization problem in (71) admits in general only an implicitly representable solution, and thus we have used the prefix "(semi-)" above. Of course, as a less tight but less involved *explicit* lower bound of the Kullback-Leibler information divergence (relative entropy) *I*(*P*A,*n*||*P*H,*n*) one can use any term of the form max <sup>n</sup> *E L*,*tan y*,*X*0,*n* , *E L*,*sec k*,*X*0,*n* , *E L*,*hor X*0,*n* o (*y* ∈ [0, ∞[, *k* ∈ N0), as well as the following

**Corollary 12.** *(a) For all* (*β*A, *β*H, *α*A, *α*H) ∈ P*SP*\P*SP,1, all initial population sizes X*<sup>0</sup> ∈ N *and all observation horizons n* ∈ N

$$I(P\_{\mathcal{A},\mathbb{II}}||P\_{\mathcal{H},\mathbb{II}}) \ge \mathbb{E}\_{X\_0,\mathbb{II}}^L \ge \tilde{E}\_{X\_0,\mathbb{II}}^L := \max\left\{ E\_{\infty,X\_0,\mathbb{II}}^{L,\text{tan}}, E\_{0,X\_0,\mathbb{II}}^{L,\text{sec}}, E\_{X\_0,\mathbb{II}}^{L,\text{hor}} \right\} \in [0, \infty] \text{ and } \tilde{E}\_{\text{max}}^{L,\text{tan}} = \tilde{E}\_{\text{max}}^{L,\text{tan}}$$

*with E L*,*hor X*0,*n defined by* (74)*, with – possibly negatively valued– finite bound component E L*,*tan* ∞,*X*0,*n* := lim*y*→<sup>∞</sup> *E L*,*tan y*,*X*0,*n , where*

$$\mathbb{E}\_{\infty,\mathcal{X}\_{0},\mathcal{U}}^{L,\text{int}} := \begin{cases} \frac{\beta\_{\mathcal{A}}\left(\log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{U}}}\right)-1\right) + \beta\_{\mathcal{H}}}{1-\beta\_{\mathcal{A}}} \cdot \left[\mathcal{X}\_{0} - \frac{a\_{\mathcal{A}}}{1-\beta\_{\mathcal{A}}}\right] \cdot \left(1 - \left(\mathcal{\beta}\_{\mathcal{A}}\right)^{n}\right) \\ \quad + \left[\frac{a\_{\mathcal{A}}\cdot\left[\beta\_{\mathcal{A}}\cdot\left(\log\left(\frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{H}}}\right)-1\right) + \beta\_{\mathcal{H}}\right]}{\beta\_{\mathcal{A}}(1-\beta\_{\mathcal{A}})} + a\_{\mathcal{A}}\left(1 - \frac{\beta\_{\mathcal{H}}}{\beta\_{\mathcal{A}}}\right) + a\_{\mathcal{H}}\left(1 - \frac{\beta\_{\mathcal{A}}}{\beta\_{\mathcal{H}}}\right)\right] \cdot n \quad \text{if } \beta\_{\mathcal{A}} \neq 1, \\\\ \left[\beta\_{\mathcal{H}} - \log\beta\_{\mathcal{H}} - 1\right] \cdot \left[\frac{a\_{\mathcal{A}}}{2} \cdot n^{2} + \left(X\_{0} + \frac{a\_{\mathcal{A}}}{2}\right) \cdot n\right] \\ \quad + \left[a\_{\mathcal{A}}\left(1 - \beta\_{\mathcal{H}}\right) + a\_{\mathcal{H}}\left(1 - \frac{1}{\beta\_{\mathcal{H}}}\right)\right] \cdot n \quad \text{if } \beta\_{\mathcal{A}} = 1, \end{cases}$$

*and –possibly negatively valued–finite bound component*

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

*For the cases* <sup>P</sup>*SP,2* ∪ P*SP,3a* ∪ P*SP,3b one gets even <sup>E</sup>*e*<sup>L</sup> <sup>X</sup>*0,*<sup>n</sup>* > 0 *for all X*<sup>0</sup> ∈ N *and all n* ∈ N*.*

#### *5.3. Applications to Bayesian Decision Making*

As explained in Section 2.5, the Kullback-Leibler information divergence fulfills

$$I(\mathsf{P}\_{\mathcal{A},\mathbb{H}}||\mathsf{P}\_{\mathcal{H},\mathbb{H}}) = \int\_0^1 \Delta \mathcal{B} \mathcal{R}\_{\widehat{\mathcal{L}\mathcal{O}}} (p\_{\mathcal{A}}^{\text{prior}}) \cdot \left(1 - p\_{\mathcal{A}}^{\text{prior}}\right)^{-1} \cdot \left(p\_{\mathcal{A}}^{\text{prior}}\right)^{-2} \mathrm{d}p\_{\mathcal{A}}^{\text{prior}}, \qquad \text{(cf. (21) with } \lambda = 1\text{)},$$

and thus can be interpreted as *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*. Hence, by combining (21) with the investigations in the previous Sections 5.1 and 5.2, we obtain exact values respectively bounds of the above-mentioned decision risk reductions. For the sake of brevity, we omit the details here.

#### **6. Explicit Closed-Form Bounds of Hellinger Integrals**

#### *6.1. Principal Approach*

Depending on the parameter constellation (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P × (R\{0, 1}), for the Hellinger integrals *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*) we have derived in Section <sup>3</sup> corresponding lower/upper bounds respectively exact values–of recursive nature– which can be obtained by choosing appropriate *p* = *p A <sup>λ</sup>* = *p <sup>A</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*), *<sup>q</sup>* <sup>=</sup> *<sup>q</sup> A <sup>λ</sup>* = *q <sup>A</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) (*<sup>A</sup>* ∈ {*E*, *<sup>L</sup>*, *<sup>U</sup>*}) and by using those together with the recursion *a* (*q*) *n n*∈N defined by (36) as well as the sequence *b* (*p*,*q*) *n n*∈N obtained from *a* (*q*) *n n*∈N by the linear transformation (38). Both sequences are "stepwise fully evaluable" but generally seem not to admit a closed-form representation in the observation horizons *n*; consequently, the time-evolution *n* 7→ *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*)–respectively the time-evolution of the corresponding recursive bounds– can generally *not be seen explicitly*. On order to avoid this *intransparency* (at the expense of losing some precision) one can approximate (36) by a recursion that allows for a closed-form representation; by the way, this will also turn out to be useful for investigations concerning diffusion limits (cf. the next Section 7).

To explain the basic underlying principle, let us first assume some *general q* ∈]0, *βλ*[ and *λ* ∈]0, 1[. With Properties <sup>1</sup> (P1) we see that the sequence *a* (*q*) *n n*∈N is strictly negative, strictly decreasing and converges to *x* (*q*) <sup>0</sup> ∈] − *βλ*, *q* − *βλ*[. Recall that this sequence is obtained by the recursive application of the function *ξ* (*q*) *λ* (*x*) := *q* · *e <sup>x</sup>* <sup>−</sup> *<sup>β</sup>λ*, through *<sup>a</sup>* (*q*) <sup>1</sup> = *ξ* (*q*) *λ* (0) = *q* − *β<sup>λ</sup>* < 0, *a* (*q*) *<sup>n</sup>* = *ξ* (*q*) *λ a* (*q*) *n*−1 = *qea* (*q*) *<sup>n</sup>*−<sup>1</sup> − *β<sup>λ</sup>* (cf. (36)). As a first step, we want to approximate *ξ* (*q*) *λ* (·) by a linear function on the interval h *x* (*q*) 0 , 0i . Due to convexity (P9), this is done by using the tangent line of *ξ* (*q*) *λ* (·) at *x* (*q*) 0

$$\mathfrak{X}\_{\lambda}^{(q),T}(\mathbf{x}) := \prescript{}{c}{c^{(q),T}} + d^{(q),T} \cdot \mathbf{x} := \prescript{}{\mathbf{x}}\_{0}^{(q)} \left(1 - q \cdot e^{\mathbf{x}\_{0}^{(q)}}\right) + q \cdot e^{\mathbf{x}\_{0}^{(q)}} \cdot \mathbf{x} \, \tag{76}$$

as a linear lower bound, and the secant line of *ξ* (*q*) *λ* (·) across its arguments 0 and *x* (*q*) 0

$$\mathfrak{L}\_{\lambda}^{(q),\mathcal{S}}(\mathbf{x}) := \mathfrak{c}^{(q),\mathcal{S}} + d^{(q),\mathcal{S}} \cdot \mathbf{x} := q - \mathfrak{\beta}\_{\lambda} + \frac{\mathfrak{x}\_{0}^{(q)} - (q - \mathfrak{\beta}\_{\lambda})}{\mathfrak{x}\_{0}^{(q)}} \cdot \mathbf{x} \, \tag{77}$$

as a linear upper bound. With the help of these functions, we can define the *linear* recursions

$$a\_0^{(q),T} := \begin{array}{c} 0 \\ \end{array} , \qquad a\_n^{(q),T} := \mathfrak{z}\_{\lambda}^{(q),T} \left( a\_{n-1}^{(q),T} \right) \; , n \in \mathbb{N} \; \tag{78}$$

$$\text{as well as} \qquad a\_0^{(q), \mathcal{S}} := \begin{array}{c} 0 \\ \end{array} , \qquad a\_n^{(q), \mathcal{S}} := \, \mathfrak{Z}\_{\lambda}^{(q), \mathcal{S}} \left( a\_{n-1}^{(q), \mathcal{S}} \right) \; , n \in \mathbb{N} . \tag{79}$$

In the following, we will refer to these sequences as the *rudimentary closed-form sequence-bounds*. Clearly, both sequences are strictly negative (on N), strictly decreasing, and one gets the sandwiching

$$a\_n^{(q),T} < |a\_n^{(q)}| \le |a\_n^{(q),S}|\tag{80}$$

for all *n* ∈ N, with equality on the right side iff *n* = 1 (where *a* (*q*) <sup>1</sup> = *q* − *β<sup>λ</sup>* < 0); moreover,

$$\lim\_{n \to \infty} a\_n^{(q), T} = \lim\_{n \to \infty} a\_n^{(q), S} = \lim\_{n \to \infty} a\_n^{(q)} = \ge\_0^{(q)}.\tag{81}$$

Furthermore, such linear recursions allow for a closed-form representation, namely

$$a\_n^{(q),\*} = \frac{c^{(q),\*}}{1 - d^{(q),\*}} \cdot \left(1 - \left(d^{(q),\*}\right)^n\right) \\ = \left. x\_0^{(q)} \cdot \left(1 - \left(d^{(q),\*}\right)^n\right)\right|,\tag{82}$$

where the " \* " stands for either *S* or *T*. Notice that this representation is valid due to *d* (*q*),*T* , *d* (*q*),*<sup>S</sup>* <sup>∈</sup>]0, 1[. So far, we have considered the case *q* ∈]0, *βλ*[. If *q* = *βλ*, then one can see from Properties 1 (P2) that *a* (*q*) *<sup>n</sup>* ≡ 0, which is also an explicitly given (though trivial) sequence. For the remaining case, where *q* > *β<sup>λ</sup>* and thus *ξ* (*q*) *λ* (0) = *a* (*q*) <sup>1</sup> <sup>=</sup> *<sup>q</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>&</sup>gt; 0), we want to exclude *<sup>q</sup>* <sup>≥</sup> min 1 , *e βλ*−1 for the following reasons. Firstly, if *q* > min 1 , *e βλ*−1 , then from (P3) we see that the sequence *a* (*q*) *n n*∈N is strictly increasing and divergent to ∞, at a rate faster than exponentially (P3b); but a linear recursion is too weak to approximate such a growth pattern. Secondly, if *q* = min 1 , *e βλ*−1 , then one necessarily gets *q* = *e <sup>β</sup>λ*−<sup>1</sup> < 1 (since we have required *q* > *βλ*, and otherwise one obtains the contradiction *β<sup>λ</sup>* < *q* = 1 ≤ *e βλ*−1 ). This means that the function *ξ* (*q*) *λ* (·) now touches the straight line *id*(·) in the point − log(*q*), i.e., *ξ* (*q*) *λ* − log(*q*) = − log(*q*). Our above-proposed method, namely to use the tangent line of *ξ* (*q*) *λ* (·) at *x* = *x* (*q*) <sup>0</sup> = − log(*q*) as a linear lower bound for *ξ* (*q*) *λ* (·), leads then to the recursion *a* (*q*),*T <sup>n</sup>* ≡ 0 (cf. (78)). This is due to the fact that the tangent line *ξ* (*q*),*T λ* (·) is in the current case equivalent with the straight line *id*(·). Consequently, (81) would not be satisfied.

Notice that in the case *β<sup>λ</sup>* < *q* < min 1 , *e βλ*−1 , the above-introduced functions *ξ* (*q*),*T λ* (·), *ξ* (*q*),*S λ* (·) constitute again linear lower and upper bounds for *ξ* (*q*) *λ* (·), however, this time on the interval <sup>h</sup> 0, *x* (*q*) 0 i . The sequences defined in (78) and (79) still fulfill the assertions (80) and (81), and additionally allow for the closed-form representation (82). Furthermore, let us mention that these rudimentary closed-form sequence-bounds can be defined analogously for *λ* ∈ R\[0, 1] and either 0 < *q* < *βλ*, or *q* = *βλ*, or max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1}.

In a second step, we want to *improve* the above-mentioned linear (lower and upper) approximations of the sequence *a* (*q*) *<sup>n</sup>* by reducing the faced error within each iteration. To do so, in both cases of lower and upper approximates we shall employ context-adapted linear *inhomogeneous difference equations* of the form

$$
\widetilde{a}\_0 := \begin{array}{c} 0 \\ \end{array}; \qquad \widetilde{a}\_n := \begin{array}{c} \widetilde{\mathfrak{s}} \left( \widetilde{a}\_{n-1} \right) \\ \end{array} + \begin{array}{c} \rho\_{n-1} & n \in \mathbb{N} \\ \end{array} \tag{83}
$$

with

$$\mathfrak{F}(\mathfrak{x}) \quad \coloneqq \quad \mathfrak{c} + d \cdot \mathfrak{x} \,, \tag{84} \\ \tag{84}$$

$$\rho\_{n-1} \quad := \quad \mathcal{K}\_1 \cdot \varkappa^{n-1} + \mathcal{K}\_2 \cdot \nu^{n-1}, \qquad n \in \mathbb{N}, \tag{85}$$

for some constants *c* ∈ R, *d* ∈]0, 1[, *K*1, *K*2,κ, *ν* ∈ R with 0 ≤ *ν* < κ ≤ *d*. This will be applied to *c* := *c* (*q*),*S* , *c* := *c* (*q*),*T* , *d* := *d* (*q*),*<sup>S</sup>* and *d* := *d* (*q*),*T* later on. Meanwhile, let us first present some facts and expressions which are insightful for further formulations and analyses.

**Lemma 2.** *Consider the sequence* (e*an*)*n*∈N<sup>0</sup> *defined in* (83) *to* (85)*. If* 0 ≤ *ν* < κ < *d, then one gets the* closed-form representation

$$
\widetilde{a}\_n = \,^\sharp \widetilde{a}\_n^{\text{hom}} + \widetilde{c}\_n \quad \text{with} \; \widetilde{a}\_n^{\text{hom}} = c \cdot \frac{1 - d^n}{1 - d} \quad \text{and} \; \widetilde{c}\_n = K\_1 \cdot \frac{d^n - \varkappa^n}{d - \varkappa} + K\_2 \cdot \frac{d^n - \nu^n}{d - \nu}, \tag{86}
$$

*which leads for all n* ∈ N *to*

$$\sum\_{k=1}^{n} \widetilde{\mathfrak{a}}\_{k} = \left( \frac{\mathcal{K}\_{1}}{d - \varkappa} + \frac{\mathcal{K}\_{2}}{d - \nu} - \frac{c}{1 - d} \right) \cdot \frac{d \cdot (1 - d^{\mathfrak{a}})}{1 - d} - \frac{\mathcal{K}\_{1} \cdot \varkappa \cdot (1 - \varkappa^{\mathfrak{a}})}{(d - \varkappa)(1 - \varkappa)} - \frac{\mathcal{K}\_{2} \cdot \nu \cdot (1 - \nu^{\mathfrak{a}})}{(d - \nu)(1 - \nu)} + \frac{c}{1 - d} \cdot \eta \cdot \mathbf{1} \tag{87}$$

*If* 0 ≤ *ν* < κ = *d, then one gets the* closed-form representation

$$
\widetilde{a}\_n = \widetilde{a}\_n^{\text{hom}} + \widetilde{c}\_n \quad \text{with} \quad \widetilde{a}\_n^{\text{hom}} = c \cdot \frac{1 - d^n}{1 - d} \quad \text{and} \quad \widetilde{c}\_n = K\_1 \cdot n \cdot d^{n-1} + K\_2 \cdot \frac{d^n - \nu^n}{d - \nu}, \tag{88}
$$

*which leads for all n* ∈ N *to*

$$\sum\_{k=1}^{n} \tilde{a}\_{k} = \left( \frac{\mathcal{K}\_{1}}{d(1-d)} + \frac{\mathcal{K}\_{2}}{d-\nu} - \frac{c}{1-d} \right) \cdot \frac{d \cdot (1-d^{u})}{1-d} - \frac{\mathcal{K}\_{2} \cdot \nu \cdot (1-\nu^{u})}{(d-\nu)(1-\nu)} + \left( \frac{c}{1-d} - \frac{\mathcal{K}\_{1} \cdot d^{u}}{1-d} \right) \cdot n \,. \tag{89}$$

Lemma 2 will be proved in Appendix A.3. Notice that (88) is consistent with taking the limit κ % *d* in (86). Furthermore, for the special case *K*<sup>2</sup> = −*K*<sup>1</sup> > 0 one has from (85) for all integers *n* ≥ 2 the relation *<sup>ρ</sup>n*−<sup>1</sup> <sup>&</sup>lt; 0 and thus <sup>e</sup>*a<sup>n</sup>* <sup>−</sup> <sup>e</sup>*<sup>a</sup> hom <sup>n</sup>* < 0, leading to

$$
\widetilde{c}\_{\mathcal{U}} < 0 \quad \text{and} \quad \sum\_{k=1}^{n} \widetilde{c}\_{\mathcal{U}} < 0 \,. \tag{90}
$$

Lemma 2 gives explicit expressions for a linear inhomogeneous recursion of the form (83) possessing the extra term given by (85). Therefrom we derive lower and upper bounds for the sequence *a* (*q*) *n n*∈N by employing *a* (*q*),*T <sup>n</sup>* resp. *a* (*q*),*S <sup>n</sup>* as the homogeneous solution of (83), i.e., by setting e*a hom n* := *a* (*q*),*T <sup>n</sup>* resp. <sup>e</sup>*<sup>a</sup> hom n* := *a* (*q*),*S <sup>n</sup>* . Moreover, our concrete approximation-error-reducing "correction terms" *ρ<sup>n</sup>* will have different form, depending on whether 0 < *q* < *β<sup>λ</sup>* or *q* > max{0, *βλ*}. In both cases, we express *ρ<sup>n</sup>* by means of the slopes *d* (*q*),*<sup>T</sup>* = *qe<sup>x</sup>* (*q*) <sup>0</sup> resp. *d* (*q*),*<sup>S</sup>* = *x* (*q*) <sup>0</sup> −(*q*−*βλ*) *x* (*q*) 0 of the tangent line *ξ* (*q*),*T λ* (·) (cf. (76)) resp. the secant line *ξ* (*q*),*S λ* (·) (cf. (77)), as well as in terms of the parameters

$$
\Gamma\_{<}^{(q)} := \frac{1}{2} \cdot \left(\mathbf{x}\_0^{(q)}\right)^2 \cdot q \cdot \varepsilon^{v\_0^{(q)}}, \quad \text{for } 0 < q < \beta\_\lambda, \qquad \text{and} \qquad \Gamma\_{>}^{(q)} := \frac{q}{2} \cdot \left(\mathbf{x}\_0^{(q)}\right)^2, \quad \text{for } q > \max\{0, \beta\_\lambda\}. \tag{91}
$$

In detail, let us first define the lower approximate by

$$\underline{a}\_{0}^{(q)} := \begin{array}{c} 0, \\ \end{array} \qquad \underline{a}\_{n}^{(q)} := \begin{array}{c} \underline{\mathfrak{z}}\_{\lambda}^{(q),T} \left( \underline{a}\_{n-1}^{(q)} \right) \\ \end{array} + \begin{array}{c} \underline{\mathfrak{z}}\_{n-1}^{(q)}, \\ \end{array} n \in \mathbb{N}, \tag{92}$$

where

$$\underline{\rho}\_{n-1}^{(q)} := \begin{cases} \Gamma\_{<}^{(q)} \cdot \left( d^{(q),T} \right)^{2(n-1)}, & \text{if } \ 0 < q < \beta\_{\lambda} \\\\ \Gamma\_{>}^{(q)} \cdot \left( d^{(q),S} \right)^{2(n-1)}, & \text{if } \ \max\{0, \beta\_{\lambda}\} < q < \min\{1, e^{\beta\_{\lambda}-1}\} \ . \end{cases} \tag{93}$$

The upper approximate is defined by

$$
\overline{\mathfrak{a}}\_0^{(q)} := \begin{array}{c} \mathbf{0} \\ \end{array} , \qquad \overline{\mathfrak{a}}\_n^{(q)} := \begin{array}{c} \mathfrak{f}\_{\lambda}^{(q), \mathcal{S}} \left( \overline{\mathfrak{a}}\_{n-1}^{(q)} \right) \\ \end{array} + \overline{\rho}\_{n-1}^{(q)} , \quad n \in \mathbb{N} \tag{94}$$

where

$$\overline{\rho}\_{n-1}^{(q)} := \begin{cases} -\Gamma\_{<}^{(q)} \cdot \left( d^{(q),T} \right)^{n-1} \cdot \left[ 1 - \left( d^{(q),S} \right)^{n-1} \right], & \text{if } 0 < q < \beta\_{\lambda} \\\\ -\Gamma\_{>}^{(q)} \cdot \left( d^{(q),S} \right)^{n-1} \cdot \left[ 1 - \left( d^{(q),T} \right)^{n-1} \right], & \text{if } \max\{0, \beta\_{\lambda} \} < q < \min\{1, e^{\beta\_{\lambda} - 1} \} \text{ .} \end{cases} \tag{95}$$

In terms of (85), we use for *ρ* (*q*) *n* the constants *K*<sup>2</sup> = *ν* = 0 as well as *K*<sup>1</sup> = Γ (*q*) <sup>&</sup>lt; , κ = *d* (*q*),*T* 2 for 0 < *q* < *β<sup>λ</sup>* respectively *K*<sup>1</sup> = Γ (*q*) <sup>&</sup>gt; , κ = *d* (*q*),*S* 2 for max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1}. For *<sup>ρ</sup>* (*q*) *<sup>n</sup>* we shall employ the constants −*K*<sup>1</sup> = *K*<sup>2</sup> = Γ (*q*) <sup>&</sup>lt; , κ = *d* (*q*),*T* , *ν* = *d* (*q*),*Sd* (*q*),*T* for 0 < *q* < *βλ*, and −*K*<sup>1</sup> = *K*<sup>2</sup> = Γ (*q*) <sup>&</sup>gt; , κ = *d* (*q*),*S* , *ν* = *d* (*q*),*Sd* (*q*),*T* for max{0, *βλ*} < *q* < min{1,*e <sup>β</sup>λ*−1}. Recall from (76) the constants *c* (*q*),*T* := *x* (*q*) 0 (<sup>1</sup> <sup>−</sup> *qe<sup>x</sup>* (*q*) <sup>0</sup> ), *d* (*q*),*T* := *qe<sup>x</sup>* (*q*) <sup>0</sup> and from (77) *c* (*q*),*S* := *q* − *βλ*, *d* (*q*),*S* := *x* (*q*) <sup>0</sup> −(*q*−*βλ*) *x* (*q*) 0 . In the following, we will refer to the sequences *a* (*q*) *<sup>n</sup>* resp. *a* (*q*) *<sup>n</sup>* as the *improved closed-form sequence-bounds*. Putting all ingredients together, we arrive at the

**Lemma 3.** *For all* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ P *there holds with d*(*q*),*<sup>T</sup>* <sup>=</sup> *qe<sup>x</sup>* (*q*) <sup>0</sup> *and d*(*q*),*<sup>S</sup>* = *x* (*q*) <sup>0</sup> −(*q*−*βλ*) *x* (*q*) 0

	- *(i)*

$$\underline{a}\_n^{(q)} < \; a\_n^{(q)} \; \le \; \overline{a}\_n^{(q)} \qquad \text{for all } n \in \mathbb{N}\_\prime$$

*with equality on the right-hand side iff n* = 1*, where*

$$\begin{split} \underline{a}\_{n}^{(q)} &= \mathbf{x}\_{0}^{(q)} \cdot \left( \mathbf{1} - \left( d^{(q),T} \right)^{n} \right) + \Gamma\_{<}^{(q)} \cdot \frac{\left( d^{(q),T} \right)^{n-1}}{1 - d^{(q),T}} \cdot \left( \mathbf{1} - \left( d^{(q),T} \right)^{n} \right) \\ \overline{a}\_{n}^{(q)} &= \mathbf{x}\_{0}^{(q)} \cdot \left( \mathbf{1} - \left( d^{(q),S} \right)^{n} \right) - \Gamma\_{<}^{(q)} \cdot \left[ \frac{\left( d^{(q),S} \right)^{n} - \left( d^{(q),T} \right)^{n}}{d^{(q),S} - d^{(q),T}} - \left( d^{(q),S} \right)^{n-1} \frac{1 - \left( d^{(q),T} \right)^{n}}{1 - d^{(q),T}} \right] \leq \mathbf{a}\_{n}^{(q),S} \end{split}$$

*with a*(*q*),*<sup>T</sup> <sup>n</sup> and a*(*q*),*<sup>S</sup> <sup>n</sup> defined by* (78) *and* (79)*.*

$$\text{(ii)}\quad \text{Both sequences } \left(\underline{a}\_n^{(q)}\right)\_{n \in \mathbb{N}} \text{ and } \left(\overline{a}\_n^{(q)}\right)\_{n \in \mathbb{N}} \text{ are strictly decreasing.}$$
 
$$\text{(iii)}$$

$$\text{The first-order coupling between the two-dimensional } \mathcal{N} \text{-matrices is the only possible } \mathcal{N} \text{-matrices with } \mathcal{N} = \{0, 1, 2, \dots, N\} \text{ and } \mathcal{N} = \{0, 1, 2, \dots, N\}.$$

$$\lim\_{n \to \infty} \underline{a}\_n^{(q)} = \lim\_{n \to \infty} \overline{a}\_n^{(q)} = \lim\_{n \to \infty} a\_n^{(q)} = \ x\_0^{(q)} \in ] - \beta\_{\lambda \prime} q - \beta\_{\lambda}[.]$$

*(b) in the case* max{0, *<sup>β</sup>λ*} <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; min 1 , *e βλ*−1 *:*

$$\langle i \rangle$$

$$\underline{a}\_{n}^{(q)} < a\_{n}^{(q)} \; \leq \; \overline{a}\_{n}^{(q)}, \qquad \text{for all } n \in \mathbb{N}\_{>}$$

*with equality on the right-hand side iff n* = 1*, where*

$$\begin{aligned} \underline{a}\_{n}^{(q)} &= \mathbf{x}\_{0}^{(q)} \cdot \left(1 - \left(d^{(q),T}\right)^{n}\right) + \Gamma\_{>}^{(q)} \cdot \frac{\left(d^{(q),T}\right)^{n} - \left(d^{(q),S}\right)^{2n}}{d^{(q),T} - \left(d^{(q),S}\right)^{2}} > a\_{n}^{(q),T} \quad \text{and} \\\ \overline{a}\_{n}^{(q)} &= \mathbf{x}\_{0}^{(q)} \cdot \left(1 - \left(d^{(q),S}\right)^{n}\right) - \Gamma\_{>}^{(q)} \cdot \left(d^{(q),S}\right)^{n-1} \left[n - \frac{1 - \left(d^{(q),T}\right)^{n}}{1 - d^{(q),T}}\right] \leq a\_{n}^{(q),S} \end{aligned}$$

*with a*(*q*),*<sup>T</sup> <sup>n</sup> and a*(*q*),*<sup>S</sup> <sup>n</sup> defined by* (78) *and* (79)*.*

*(ii) Both sequences a* (*q*) *n n*∈N *and a* (*q*) *n n*∈N *are strictly increasing. (iii)* lim*n*→<sup>∞</sup> *a* (*q*) *<sup>n</sup>* <sup>=</sup> lim*n*→<sup>∞</sup> *a* (*q*) *<sup>n</sup>* <sup>=</sup> lim*n*→<sup>∞</sup> *a* (*q*) *<sup>n</sup>* = *x* (*q*) <sup>0</sup> ∈]*q* − *βλ*, − log(*q*)[.

A detailed proof of Lemma 3 is provided in Appendix A.3. In the following, we employ the above-mentioned investigations in order to derive the desired closed-form bounds of the Hellinger integrals *Hλ*(*P*A,*n*||*P*H,*n*).

*(b)*

*6.2. Explicit Closed-Form Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*NI* ∪ P*SP,1*) × (R\{0, 1})

Recall that in this setup, we have obtained the recursive, non-explicit *exact* values *Vλ*,*X*0,*<sup>n</sup>* = *Hλ*(*P*A,*n*||*P*H,*n*) given in (39) of Theorem 1, where we used *q* = *q E <sup>λ</sup>* = *q E* (*β*A, *β*H, *λ*) = *β λ* A *β* 1−*λ* H ∈]0, *βλ*[ in the case *λ* ∈]0, 1[ respectively *q* = *q E <sup>λ</sup>* = *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; max{0, *<sup>β</sup>λ*} in the case *<sup>λ</sup>* <sup>∈</sup> <sup>R</sup>\[0, 1]. For the latter, Lemma 1 implies that *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]. This—together with (39) from Theorem 1, Lemma 2 and with the quantities *d* (*q*),*T* , *d* (*q*),*S* , Γ (*q*) <sup>&</sup>lt; and Γ (*q*) <sup>&</sup>gt; as defined in (76) and (77) resp. (91) –leads to

**Theorem 5.** *Let p E λ* := *α λ* A *α* 1−*λ* H *and q E λ* := *β λ* A *β* 1−*λ* H *. For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*NI* ∪ P*SP,1*) × ]*λ*−, *λ*+[ \ {0, 1} *, all initial population sizes X*<sup>0</sup> ∈ N *and for all observation horizons n* ∈ N *the following assertions hold true:*

*(a) the Hellinger integral can be bounded by the closed-form lower and upper bounds*

*C* (*p E λ* ,*q E λ* ),*T <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≤ *C* (*p E λ* ,*q E λ* ),*L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≤ *Vλ*,*X*0,*<sup>n</sup>* = *Hλ*(*P*A,*n*||*P*H,*n*) ≤ *C* (*p E λ* ,*q E λ* ),*U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≤ *C* (*p E λ* ,*q E λ* ),*S λ*,*X*0,*n* , lim*n*→<sup>∞</sup> 1 *n* log *Vλ*,*X*0,*<sup>n</sup>* <sup>=</sup> lim*n*→<sup>∞</sup> 1 *n* log *C* (*p E λ* ,*q E λ* ),*L λ*,*X*0,*n* <sup>=</sup> lim*n*→<sup>∞</sup> 1 *n* log *C* (*p E λ* ,*q E λ* ),*U λ*,*X*0,*n* <sup>=</sup> lim*n*→<sup>∞</sup> 1 *n* log *C* (*p E λ* ,*q E λ* ),*T λ*,*X*0,*n* <sup>=</sup> lim*n*→<sup>∞</sup> 1 *n* log *C* (*p E λ* ,*q E λ* ),*S λ*,*X*0,*n* = *α*A *β*A · *x* (*q E λ* ) 0

*where the involved closed-form lower bounds are defined by*

$$\begin{array}{rcl}\mathbf{C}\_{\lambda,\mathbf{X}\_{0},\mu}^{(p\_{\mathcal{E}}^{\mathbb{E}}\underline{\boldsymbol{q}}\_{\lambda}^{\mathbb{E}}),\boldsymbol{\mathcal{I}}} :=& \mathbf{C}\_{\lambda,\mathbf{X}\_{0},\mu}^{(p\_{\mathcal{E}}^{\mathbb{E}}\underline{\boldsymbol{q}}\_{\lambda}^{\mathbb{E}}),\boldsymbol{\mathcal{T}}} \cdot \exp\left\{\underline{\boldsymbol{\zeta}}\_{\mu}^{(q\_{\mathcal{E}}^{\mathbb{E}})} \cdot \mathbf{X}\_{0} + \frac{\boldsymbol{\mathfrak{a}}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \cdot \underline{\boldsymbol{\mathfrak{a}}}\_{\mu}^{(q\_{\mathcal{E}}^{\mathbb{E}})} \right\}, & \text{ with} \\\mathbf{C}\_{\lambda,\mathbf{X}\_{0},\mu}^{(p\_{\mathcal{E}}^{\mathbb{E}}\underline{\boldsymbol{q}}\_{\lambda}^{\mathbb{E}}),\boldsymbol{\mathcal{T}}} :=& \exp\left\{\mathbf{x}\_{0}^{(q\_{\mathcal{E}}^{\mathbb{E}})} \cdot \left[\mathbf{X}\_{0} - \frac{\boldsymbol{\mathfrak{a}}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \cdot \frac{d^{(q\_{\mathcal{E}}^{\mathbb{E}}),\boldsymbol{\mathcal{T}}}}{1 - d^{(q\_{\mathcal{E}}^{\mathbb{E}}),\boldsymbol{\mathcal{T}}}} \right] \cdot \left(1 - \left(d^{(q\_{\mathcal{E}}^{\mathbb{E}}),\boldsymbol{\mathcal{T}}}\right)^{\mathrm{H}}\right) + \frac{\boldsymbol{\mathfrak{a}}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \mathbf{x}\_{0}^{(q\_{\mathcal{E}}^{\mathbb{E}})} \cdot n \right\}, \end{array}$$

,

*and the closed-form upper bounds are defined by*

$$\begin{array}{lcl}\mathbf{C}^{(p\_{\mathcal{A}}^{\mathbb{F}}\underline{\boldsymbol{\mu}}\_{\lambda}^{\mathbb{F}})\boldsymbol{\Lambda}} & := & \mathbf{C}^{(p\_{\mathcal{A}}^{\mathbb{F}}\underline{\boldsymbol{\mu}}\_{\lambda}^{\mathbb{F}}),\mathbf{S}} \cdot \exp\left\{-\widetilde{\boldsymbol{\zeta}}\_{\boldsymbol{n}}^{(q\_{\lambda}^{\mathbb{F}})} \cdot \mathbf{X}\_{0} - \frac{\boldsymbol{\mathsf{a}}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \cdot \overline{\boldsymbol{\theta}}\_{\boldsymbol{n}}^{(q\_{\lambda}^{\mathbb{F}})} \right\}, & \text{with} \\ \mathbf{C}^{(p\_{\mathcal{A}}^{\mathbb{F}}\underline{\boldsymbol{\mu}}\_{\lambda}^{\mathbb{F}}),\mathbf{S}} & := & \exp\left\{\mathbf{x}\_{0}^{(q\_{\lambda}^{\mathbb{F}})} \cdot \left[\mathbf{X}\_{0} - \frac{\boldsymbol{\mathsf{a}}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \cdot \frac{d^{(q\_{\lambda}^{\mathbb{F}}),\mathbf{S}}}{1 - d^{(q\_{\lambda}^{\mathbb{F}}),\mathbf{S}}}\right] \cdot \left(1 - \left(d^{(q\_{\lambda}^{\mathbb{F}}),\mathbf{S}}\right)^{\mathbf{n}}\right) + \frac{\boldsymbol{\mathsf{a}}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{A}}} \mathbf{x}\_{0}^{(q\_{\lambda}^{\mathbb{F}})} \cdot n\right), \end{array}$$

*where in the case λ* ∈]0, 1[

$$\underline{\mathcal{L}}\_{\underline{n}}^{(q\_{\underline{k}}^{\mathbb{E}})} := \; \Gamma\_{\underline{<}}^{(q\_{\underline{k}}^{\mathbb{E}})} \cdot \frac{\left(d^{(q\_{\underline{k}}^{\mathbb{E}}), \Gamma}\right)^{\mathfrak{n}-1}}{1 - d^{(q\_{\underline{k}}^{\mathbb{E}}), \Gamma}} \cdot \left(1 - \left(d^{(q\_{\underline{k}}^{\mathbb{E}}), \Gamma}\right)^{\mathfrak{n}}\right) > 0 \,, \tag{98}$$

$$\underline{\theta}\_{n}^{\{q\_{\lambda}^{\mathbb{E}}\}} := \quad \Gamma\_{<}^{\{q\_{\lambda}^{\mathbb{E}}\}} \cdot \frac{1 - \left(d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{n}}{\left(1 - d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{2}} \cdot \left[1 - \frac{d^{(q\_{\lambda}^{\mathbb{E}}),T} \left(1 + \left(d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{n}\right)}{1 + d^{(q\_{\lambda}^{\mathbb{E}}),T}}\right] \\ > 0 \,,\tag{99}$$

$$\overline{\xi}\_{n}^{\{q\_{\lambda}^{\mathbb{E}}\}} := \ \Gamma\_{<}^{\{q\_{\lambda}^{\mathbb{E}}\}} \cdot \left[ \frac{\left(d^{(q\_{\lambda}^{\mathbb{E}}),S}\right)^{n} - \left(d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{n}}{d^{(q\_{\lambda}^{\mathbb{E}}),S} - d^{(q\_{\lambda}^{\mathbb{E}}),T}} - \left(d^{(q\_{\lambda}^{\mathbb{E}}),S}\right)^{n-1} \cdot \frac{1 - \left(d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{n}}{1 - d^{(q\_{\lambda}^{\mathbb{E}}),T}} \right] \\ > 0 \,, \tag{100}$$

$$\overline{\theta}\_{n}^{(q\_{\lambda}^{\mathbb{E}})} := \quad \Gamma\_{<}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \frac{d^{(q\_{\lambda}^{\mathbb{E}}),T}}{1 - d^{(q\_{\lambda}^{\mathbb{E}}),T}} \cdot \left[ \frac{1 - \left(d^{(q\_{\lambda}^{\mathbb{E}}),S}d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{\mathbb{H}}}{1 - d^{(q\_{\lambda}^{\mathbb{E}}),S}d^{(q\_{\lambda}^{\mathbb{E}}),T}} - \frac{\left(d^{(q\_{\lambda}^{\mathbb{E}}),S}\right)^{\mathbb{H}} - \left(d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{\mathbb{H}}}{d^{(q\_{\lambda}^{\mathbb{E}}),S} - d^{(q\_{\lambda}^{\mathbb{E}}),T}} \right] > 0 \,, \tag{101}$$

*and where in the case λ* ∈ ]*λ*−, *λ*+[ \[0, 1]

$$\underline{\mathcal{L}}\_{\rm{u}}^{(q\_{\underline{\lambda}}^{\mathbb{E}})} := \quad \Gamma\_{>}^{(q\_{\underline{\lambda}}^{\mathbb{E}})} \cdot \frac{\left(d^{(q\_{\underline{\lambda}}^{\mathbb{E}})}\_{\lambda}\right)^{n} - \left(d^{(q\_{\underline{\lambda}}^{\mathbb{E}})}\_{\lambda}\right)^{2n}}{d^{(q\_{\underline{\lambda}}^{\mathbb{E}}),T} - \left(d^{(q\_{\underline{\lambda}}^{\mathbb{E}})}\_{\lambda}\right)^{2}} > 0 \,, \tag{102}$$
 
$$\Gamma\_{\leq}^{(q\_{\underline{\lambda}}^{\mathbb{E}})} \qquad \left[d^{(q\_{\underline{\lambda}}^{\mathbb{E}}),T} \left(1 - \left(d^{(q\_{\underline{\lambda}}^{\mathbb{E}}),T}\_{\lambda}\right)^{n}\right) \quad \left(d^{(q\_{\underline{\lambda}}^{\mathbb{E}}),S}\right)^{2} \left(1 - \left(d^{(q\_{\underline{\lambda}}^{\mathbb{E}}),S}\_{\lambda}\right)^{2n}\right)\right]$$

$$\underline{d}\_{n}^{(q\_{\lambda}^{\mathbb{E}})} := \frac{\Gamma\_{>}^{(q\_{\lambda}^{\mathbb{E}})}}{d^{(q\_{\lambda}^{\mathbb{E}}),T} - \left(d^{(q\_{\lambda}^{\mathbb{E}}),S}\right)^{2}} \left[\frac{d^{(q\_{\lambda}^{\mathbb{E}}),T} \left(1 - \left(d^{(q\_{\lambda}^{\mathbb{E}}),T}\right)^{\prime\prime}\right)}{1 - d^{(q\_{\lambda}^{\mathbb{E}}),T}} - \frac{\left(d^{(q\_{\lambda}^{\mathbb{E}}),\rhd}\right) \left(1 - \left(d^{(q\_{\lambda}^{\mathbb{E}}),\rhd}\right)\right)}{1 - \left(d^{(q\_{\lambda}^{\mathbb{E}}),S}\right)^{2}}\right] \tag{103}$$

$$\nabla\_{\mathbf{n}}^{(q\_{\lambda}^{\mathbb{E}})} := \quad \Gamma\_{>}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \left( d^{(q\_{\lambda}^{\mathbb{E}}),S} \right)^{n-1} \cdot \left[ n - \frac{1 - \left( d^{(q\_{\lambda}^{\mathbb{E}}),T} \right)^{n}}{1 - d^{(q\_{\lambda}^{\mathbb{E}}),T}} \right] \\ > 0 \,, \tag{104}$$

$$\begin{split} \overline{\theta}\_{n}^{(q\_{\overline{\lambda}}^{\mathbb{E}})} &:= \quad \Gamma\_{>}^{(q\_{\overline{\lambda}}^{\mathbb{E}})} \cdot \left[ \frac{d^{(q\_{\overline{\lambda}}^{\mathbb{E}})} \mathcal{S} - d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), T}}{\left(1 - d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), \mathcal{S}}\right)^{2} \left(1 - d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), T}\right)} \cdot \left(1 - \left(d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), \mathcal{S}}\right)^{n}\right) \\ &+ \frac{d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), T} \left(1 - \left(d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), S} d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), T}\right)^{n}\right)}{\left(1 - d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), T}\right) \left(1 - d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), S} d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), T}\right)} - \frac{\left(d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), S}\right)^{n}}{1 - d^{(q\_{\overline{\lambda}}^{\mathbb{E}}), S}} \cdot n \right] > 0 \,. \end{split} \tag{105}$$

*Notice that <sup>α</sup>*<sup>A</sup> *β*A *can be equivalently be replaced by <sup>α</sup>*<sup>H</sup> *β*H *in* (96) *and in* (97)*.*

A proof of Theorem 5 is given in Appendix A.3.
