*3.11. Upper Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4a*×]0, 1[

This setup and the remaining setup (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,4b×]0, 1[ (see the next Section 3.12) are the only constellations where *φλ*(·) is strictly negative and strictly increasing, with lim*x*→<sup>∞</sup> *φλ*(*x*) = lim*x*→<sup>∞</sup> *φ* 0 *λ* (*x*) = 0, leading to the choices *p U <sup>λ</sup>* = *α<sup>λ</sup>* as well as *q U <sup>λ</sup>* = *β<sup>λ</sup>* = *β* under the restriction that exp <sup>n</sup> *a* (*q U λ* ) *<sup>n</sup>* · *X*<sup>0</sup> + ∑ *n k*=1 *b* (*p U λ* ,*q U λ* ) *k* o ≤ 1 for all *n* ∈ N and all *X*<sup>0</sup> ∈ N. Consequently, one has *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1, which is consistent with the general upper bound (9) but violates the above-mentioned desired Goal (G1). Unfortunately, the proof method of (51) (cf. Appendix A.1) can't be carried over to the current setup. The following proposition states two of the above-mentioned desired assertions which can be verified by a completely different proof method, which is also given in Appendix A.1.

**Proposition 9.** *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 *the parts (c) and (d) of Proposition 8 hold true.*

*3.12. Upper Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4b*×]0, 1[

The assertions preceding Proposition 9 remain valid. However, any linear upper bound of the function *φλ*(·) on the domain N<sup>0</sup> possesses the slope *q U <sup>λ</sup>* − *β<sup>λ</sup>* ≥ 0. If *q U <sup>λ</sup>* = *βλ*, then the intercept is *p U <sup>λ</sup>* − *α<sup>λ</sup>* = 0 leading to *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1 and thus Goal (G1) is violated. If we use a slope *q U <sup>λ</sup>* − *β<sup>λ</sup>* > 0, then both the sequences *a* (*q U λ* ) *n n*∈N and *b* (*p U λ* ,*q U λ* ) *n n*∈N are strictly increasing and diverge to ∞. This comes from Properties 1 (P3b) and (P7b) since *q U <sup>λ</sup>* > *β<sup>λ</sup>* = *β* ≥ 1. Altogether, this implies that the corresponding upper bound component *B*e (*p U λ* ,*q U λ* ) *λ*,*X*0,*n* (cf. (42)) diverges to ∞ as well. This leads to

**Proposition 10.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4b*×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there do not exist parameters p U <sup>λ</sup>* ≥ 0*, q U <sup>λ</sup>* ≥ 0 *such that* (35) *is satisfied and such that the parts (c) and (d) of Proposition 8 hold true.*

*3.13. Concluding Remarks on Alternative Upper Bounds for all Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[

As mentioned earlier on, starting from Section 3.6 we have principally focused on constructing upper bounds *B U λ*,*X*0,*n* of the Hellinger integrals, starting from *p U λ* , *q U <sup>λ</sup>* which fulfill (35) as well as further constraints depending on the Goals (G1) and (G2). For the setups in the Sections 3.7–3.9, we have

proved the existence of *special parameter choices p U λ* , *q U <sup>λ</sup>* which were consistent with (G1) and (G2). Furthermore, for the constellation in the Section 3.11 we have found parameters such that at least (G2) is satisfied. In contrast, for the setup of Section 3.12 we have not found any choices which are consistent with (G1) and (G2), leading to the "cut-off bound" *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ <sup>1</sup> which gives no improvement over the generally valid upper bound (9).

In the following, we present some *alternative choices* of *p U λ* , *q U <sup>λ</sup>* which–depending on the parameter constellation (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (PSP\PSP,1)×]0, 1[–may or may not lead to upper bounds *B U λ*,*X*0,*n* which are consistent with Goal (G1) or with (G2) (and which are maybe weaker or better than resp. incomparable with the previous upper bounds when dealing with some relaxations of (G1), such as e.g., *Hλ*(*P*A,*n*||*P*H,*n*) < 1 for all but finitely many *n* ∈ N).

As a first alternative choice for a linear upper bound of *φλ*(·) (cf. (35)) one could use the asymptote *φ*f*λ*(·) (cf. Properties 3 (P20)) with the parameters *p U λ* :<sup>=</sup> <sup>f</sup>*p<sup>λ</sup>* <sup>=</sup> *λα*<sup>A</sup> (*β*A/*β*H) *<sup>λ</sup>*−<sup>1</sup> + (<sup>1</sup> <sup>−</sup> *λ*)*α*<sup>H</sup> (*β*A/*β*H) *λ* and *q U λ* :<sup>=</sup> *<sup>q</sup>*e*<sup>λ</sup>* <sup>=</sup> *<sup>β</sup> λ* A *β* 1−*λ* H . Another important linear upper bound of *φλ*(·) is the tangent line *φ* tan *λ*,*y* (·) on *φλ*(·) at an arbitrarily fixed point *y* ∈ [0, ∞[, which amounts to

$$\boldsymbol{\phi}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\tan}(\mathbf{x}) := \boldsymbol{r}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\tan} + \boldsymbol{s}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\tan} \cdot \mathbf{x} := \left(\boldsymbol{p}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\tan} - \boldsymbol{a}\_{\boldsymbol{\lambda}}\right) + \left(\boldsymbol{q}\_{\boldsymbol{\lambda},\boldsymbol{y}}^{\tan} - \boldsymbol{\beta}\_{\boldsymbol{\lambda}}\right) \cdot \mathbf{x} := \left(\boldsymbol{\phi}\_{\boldsymbol{\lambda}}(\boldsymbol{y}) - \boldsymbol{y} \cdot \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y})\right) + \boldsymbol{\phi}\_{\boldsymbol{\lambda}}'(\boldsymbol{y}) \cdot \mathbf{x} \tag{52}$$

where *φ* 0 *λ* (·) is given by (P17). Notice that this upper bound is for *y* ∈]0, ∞[\N "not tight" in the sense that *φ* tan *λ*,*y* (·) does not hit the function *φλ*(·) on N<sup>0</sup> (where the generation sizes "live"); moreover, *φ* tan *λ*,*y* (*x*) might take on strictly positive values for large enough points *x* which is counter-productive for Goal (G1). Another alternative choice of a linear upper bound for *φλ*(·), which in contrast to the tangent line is "tight" (but not necessarily avoiding the strict positivity), is the secant line *φ* sec *λ*,*k* (·) across its arguments *k* and *k* + 1, given by

$$\begin{array}{rcl}\phi\_{\lambda,k}^{\mathrm{sec}}(\mathbf{x}) & \ := & r\_{\lambda,k}^{\mathrm{sec}} + s\_{\lambda,k}^{\mathrm{sec}} \cdot \mathbf{x} \ := & \left(p\_{\lambda,k}^{\mathrm{sec}} - \mathbf{a}\_{\lambda}\right) + \left(q\_{\lambda,k}^{\mathrm{sec}} - \beta\_{\lambda}\right) \cdot \mathbf{x} \\ & \ := & \left[\phi\_{\lambda}(k) - k \cdot \left(\phi\_{\lambda}(k+1) - \phi\_{\lambda}(k)\right)\right] \ + \left(\phi\_{\lambda}(k+1) - \phi\_{\lambda}(k)\right) \cdot \mathbf{x} \ .\end{array} \tag{53}$$

Another alternative choice is the horizontal line

$$\phi\_{\lambda}^{\text{hor}}(\mathbf{x}) \equiv \max \left\{ \phi\_{\lambda}(y), \, y \in \mathbb{N}\_{0} \right\}. \tag{54}$$

For *p U λ* ∈ n <sup>f</sup>*p<sup>λ</sup>* , *<sup>p</sup>* tan *λ*,*y* , *p* sec *λ*,*y* o and *q U λ* ∈ n *q* tan *λ*,*y* , *q* sec *λ*,*y* o it is possible that in some parameter cases (*β*A, *β*H, *α*A, *α*H) either the intercept *r U <sup>λ</sup>* = *p U <sup>λ</sup>* − *α<sup>λ</sup>* is strictly larger than zero or the slope *s U <sup>λ</sup>* = *q U <sup>λ</sup>* − *β<sup>λ</sup>* is strictly larger than zero. Thus, it can happen that *B*e (*p U λ* ,*q U λ* ) *<sup>λ</sup>*,*X*0,*<sup>n</sup>* > 1 for some (and even for all) *n* ∈ N, such that the corresponding upper bound *B U λ*,*X*0,*n* for the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*) amounts to the cut-off at 1. However, due to Properties <sup>1</sup> (P5) and (P7a), the sequence *B*e (*p U λ* ,*q U λ* ) *λ*,*X*0,*n n*∈N may become smaller than 1 and may finally converge to zero. Due to Properties 2 (P14), this upper bound can even be tighter (smaller) than those bounds derived from parameters *p U λ* , *q U λ* fulfilling (47).

As far as our desired Hellinger integral bounds are concerned, in the setup of Section 3.11 —where lim*y*→<sup>∞</sup> *φ* tan *λ*,*y* (·) ≡ 0–for the proof of Proposition 9 in Appendix A.1 we shall employ the mappings *y* 7→ *φ* tan *λ*,*y* resp. *y* 7→ *p* tan *λ*,*y* resp. *y* 7→ *q* tan *λ*,*y* . These will also be used for the proof of the below-mentioned Theorem 4.

#### *3.14. Intermezzo 1: Application to Asymptotical Distinguishability*

The above-mentioned investigations can be applied to the context of Section 2.6 on asymptotical distinguishability. Indeed, with the help of the Definitions 1 and 2 as well as the equivalence relations (25) and (26) we obtain the following

#### **Corollary 1.**


The proof of Corollary 1 will be given in Appendix A.1.

#### **Remark 3.**


(*P*A,*n*) / .(*P*H,*n*) *if β*<sup>A</sup> ≤ 1*, β*<sup>H</sup> ≤ 1*;* (*P*A,*n*) / . (*P*H,*n*) *if β*<sup>A</sup> ≤ 1*, β*<sup>H</sup> > 1*;* (*P*A,*n*) / . (*P*H,*n*) *if β*<sup>A</sup> > 1*, β*<sup>H</sup> ≤ 1*;* (*P*A,*n*) / . (*P*H,*n*) *and* (*P*A,*n*)4(*P*H,*n*) *if β*<sup>A</sup> > 1*, β*<sup>H</sup> > 1*; in particular, for* P*NI the sequences* (*P*A,*n*)*n*∈N<sup>0</sup> *and* (*P*H,*n*)*n*∈N<sup>0</sup> *are not completely asymptotically inseparable (indistinguishable).*


#### 3.15.1. Bayesian Decision Making

The above-mentioned investigations can be applied to the context of Section 2.5 on *dichotomous* Bayesian 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 upper (respectively lower) bounds by using (19) respectively (20) together with the exact values or the upper (respectively lower) bounds of the Hellinger integrals *Hλ*(*P*A,*n*||*P*H,*n*) derived in the "*λ* ∈]0, 1[ parts" of Theorem 1, the Sections 3.3–3.13 (and also in the below-mentioned Section 6); instead of providing the corresponding outcoming formulas–which is merely repetitive–we give the illustrative

**Example 1.** *Based on a sample path observation* X*<sup>n</sup>* := {*X*` : ` = 1, ..., *n*} *of a GWI, which is either governed by a hypothesis law P*<sup>H</sup> *or an alternative law P*A*, we want to make a dichotomous optimal Bayesian decision described in Section 2.5, namely, decide between an action d*<sup>H</sup> *"associated with" P*<sup>H</sup> *and an action d*<sup>A</sup> *"associated with" P*A*, with pregiven loss function* (16) *involving constants L*<sup>A</sup> > 0*, L*<sup>H</sup> > 0 *which e.g., arise as bounds from quantities in worst-case scenarios.*

*For this, let us exemplarily deal with initial population X*<sup>0</sup> = 5 *as well as parameter setup* (*β*A, *β*H, *α*A, *α*H) = (1.2, 0.9, 4, 3) ∈ P*SP,1; within our running-example epidemiological context of Section 2.3, this corresponds e.g., to a setup where one is encountered with a novel infectious disease (such as COVID-19) of non-negligible fatality rate, and* (A) *reflects a "potentially dangerous" infectious-disease-transmission situation (with supercritical reproduction number β*<sup>A</sup> = 1.2 *and importation mean of α*<sup>A</sup> = 4*, for weekly appearing new incidence-generations) whereas* (H) *describes a "milder" situation (with subcritical β*<sup>H</sup> = 0.9

*and α*<sup>H</sup> = 3*). Moreover, let d*<sup>H</sup> *and d*<sup>A</sup> *reflect two possible sets of interventions (control measures) in the course of pandemic risk management, with respective "worst-case type" decision losses L*<sup>A</sup> = 600 *and L*<sup>H</sup> = 300 *(e.g., in units of billion Euros or U.S. Dollars). Additionally we assume the prior probabilities π* = *Pr*(H) = 1 − *Pr*(A) = 0.5*, which results in the prior-loss constants* L<sup>A</sup> = 300 *and* L<sup>H</sup> = 150*. In order to obtain bounds for the corresponding minimal mean decision loss (Bayes Risk)* R*<sup>n</sup> defined in* (18) *we can employ the general Stummer-Vajda bounds (cf. [15])* (19) *and* (20) *in terms of the Hellinger integral Hλ*(*P*A,*n*||*P*H,*n*) *(with arbitrary λ* ∈]0, 1[*), and combine this with the appropriate detailed results on the latter from the preceding subsections. To demonstrate this, let us choose λ* = 0.5 *(for which H*1/2(*P*A,*n*||*P*H,*n*) *can be interpreted as a multiple of the Bhattacharyya coefficient between the two competing GWI) respectively λ* = 0.9*, leading to the parameters p E* 0.5 = 3.464, *q E* 0.5 = 1.039 *respectively p E* 0.9 = 3.887*, q E* 0.9 = 1.166 *(cf.* (33)*). Combining* (19) *and* (20) *with Theorem 1 (a)– which provides us with the exact recursive values of <sup>H</sup>λ*(*P*A,*n*||*P*H,*n*) *in terms of the sequence a*(*<sup>q</sup> E λ* ) *<sup>n</sup> (cf.* (36)*)– we obtain for λ* = 0.5 *the bounds*

$$\begin{aligned} \mathcal{R}\_n &\leq \quad \mathcal{R}\_n^{\mathrm{II}} := 2.121 \cdot 10^2 \cdot \exp\left\{ 5 \cdot a\_n^{(1.039)} + \frac{10}{3} \cdot \sum\_{k=1}^n a\_k^{(1.039)} \right\}, \\\mathcal{R}\_n &\geq \quad \mathcal{R}\_n^{\mathrm{L}} := 100 \cdot \exp\left\{ 10 \cdot a\_n^{(1.039)} + \frac{20}{3} \cdot \sum\_{k=1}^n a\_k^{(1.039)} \right\}. \end{aligned}$$

*whereas for λ* = 0.9 *we get*

$$\begin{aligned} \mathcal{R}\_{\boldsymbol{n}} & \leq \quad \mathcal{R}\_{\boldsymbol{n}}^{\boldsymbol{L}} := 2.799 \cdot 10^{2} \cdot \exp\left\{ 5 \cdot a\_{\boldsymbol{n}}^{(1.166)} + \frac{10}{3} \cdot \sum\_{k=1}^{\boldsymbol{n}} a\_{\boldsymbol{k}}^{(1.166)} \right\}, \\ \mathcal{R}\_{\boldsymbol{n}} & \geq \quad \mathcal{R}\_{\boldsymbol{n}}^{\boldsymbol{L}} := 3.902 \cdot \exp\left\{ 50 \cdot a\_{\boldsymbol{n}}^{(1.166)} + \frac{100}{3} \cdot \sum\_{k=1}^{\boldsymbol{n}} a\_{\boldsymbol{k}}^{(1.166)} \right\}. \end{aligned}$$

*Figure <sup>1</sup> illustrates the lower (orange resp. cyan) and upper (red resp. blue) bounds* <sup>R</sup>*<sup>L</sup> n resp.* <sup>R</sup>*<sup>U</sup> n of the Bayes Risk* R*<sup>n</sup> employing λ* = 0.5 *resp. λ* = 0.9 *on both a unit scale (left graph) and a logarithmic scale (right graph). The lightgrey/grey/black curves correspond to the* (18)*-based empirical evaluation of the Bayes risk sequence* R *sample n <sup>n</sup>*=1,...,50 *from three independent Monte Carlo simulations of 10000 GWI sample paths (each) up to time horizon 50.*

**Figure 1.** Bayes risk bounds (using *λ* = 0.5 (red/orange) resp. *λ* = 0.9 (blue/cyan)) and Bayes risk simulations (lightgrey/grey/black) on a unit (**left graph**) and logarithmic (**right graph**) scale in the parameter setup (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) = (1.2, 0.9, 4, 3) ∈ PSP,1, with initial population *<sup>X</sup>*<sup>0</sup> = 5 and prior-loss constants <sup>L</sup><sup>A</sup> = 300 and <sup>L</sup><sup>H</sup> = 150.

#### 3.15.2. Neyman-Pearson Testing

By combining (23) with the exact values resp. upper bounds of the Hellinger integrals *H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*) from the preceding subsections, we obtain for our context of GW(I) with Poisson offspring and Poisson immigration (including the non-immigration case) some upper bounds of the *minimal* type II error probability E*<sup>ς</sup>* (*P*A,*n*||*P*H,*n*) in the class of the tests for which the type I error probability is at most *ς* ∈]0, 1[, which can also be immediately rewritten as lower bounds for the power 1 − E*<sup>ς</sup>* (*P*A,*n*||*P*H,*n*) of a most powerful test at level *ς*. As for the Bayesian context of Section 3.15.1, instead of providing the–merely repetitive–outcoming formulas for the bounds of E*<sup>ς</sup>* (*P*A,*n*||*P*H,*n*) we give the illustrative

**Example 2.** *Consider the Figures 2 and 3 which deal with initial population X*<sup>0</sup> = 5 *and the parameter setup* (*β*A, *β*H, *α*A, *α*H) = (0.3, 1.2, 1, 4) ∈ P*SP,1; within our running-example epidemiological context of Section 2.3, this corresponds to a "potentially dangerous" infectious-disease-transmission situation* (H) *(with supercritical reproduction number β*<sup>H</sup> = 1.2 *and importation mean of α*<sup>H</sup> = <sup>4</sup>*), whereas* (A) *describes a "very mild" situation (with "low" subcritical β*<sup>A</sup> = 0.3 *and α*<sup>A</sup> = 1*). Figure 2 shows the lower and upper bounds of* E*<sup>ς</sup>* (*P*A,*n*||*P*H,*n*) *with ς* = 0.05*, evaluated from the Formulas* (23) *and* (24)*, together with the exact values of the Hellinger integral H<sup>λ</sup>* (*P*A,*n*||*P*H,*n*)*, cf. Theorem 1 (recall that we are in the setup* P*SP,1) on both a unit scale (left graph) and a logarithmic scale (right graph). The orange resp. red resp. purple curves correspond to the outcoming upper bounds* E *U n* := E *U n* (*P*A,*n*||*P*H,*n*) *(cf.* (23)*) with parameters λ* = 0.3 *resp. λ* = 0.5 *resp. λ* = 0.7*. The green resp. cyan resp. blue curves correspond to the lower bounds* E *L n* := E *L n* (*P*A,*n*||*P*H,*n*) *(cf.* (24)*) with parameters λ* = 2 *resp. λ* = 1.5 *resp. λ* = 1.1*. Notice the different λ-ranges in* (23) *and* (24)*. In contrast, Figure 3 compares the lower bound* E *L n (for fixed λ* = 1.1*) with the upper bound* E *U n (for fixed λ* = 0.5*) of the minimal type II error probability* E*ς*(*P*A,*n*||*P*H,*n*) *for different levels ς* = 0.1 *(orange for the lower and cyan for the upper bound), ς* = 0.05 *(green and magenta) and ς* = 0.01 *(blue and purple) on both a unit scale (left graph) and a logarithmic scale (right graph).*

**Figure 2.** Different lower bounds E *L n* using *λ* ∈ {1.1, 1.5, 2} and upper bounds E *U n* using *λ* ∈ {0.3, 0.5, 0.7} of the minimal type II error probability E*<sup>ς</sup>* (*P*A,*n*||*P*H,*n*) for fixed level *<sup>ς</sup>* = 0.05 in the parameter setup (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) = (0.3, 1.2, 1, 4) ∈ PSP,1 together with initial population *<sup>X</sup>*<sup>0</sup> = 5 on both a unit scale (**left graph**) and a logarithmic scale (**right graph**).

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

Recall from (49) the set PSP := (*β*A, *β*H, *α*A, *α*H) ∈ ]0, ∞[ 4 : (*α*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>α</sup>*H) or (*β*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*H) or both and the "equal-fraction-case" set PSP,1 := n (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> 6= *α*H, *β*<sup>A</sup> 6= *β*H, *α*A *β*A = *α*H *β*H o , where for the latter we have derived in Theorem 1(a) and in Proposition 5 the *exact* recursive values for

the time-behaviour of the Hellinger integrals *Hλ*(*P*A,1||*P*H,1) of order *λ* ∈ R\[0, 1]. Moreover, recall that for the case (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (PSP\PSP,1)×]0, 1[ we have obtained in the Sections 3.4 and 3.5 some "optimal" linear lower bounds *φ L λ* (·) for the strictly concave function *φλ*(*x*) := *φ*(*x*, *β*A, *β*H, *α*A, *α*H, *λ*) on the domain *x* ∈ [0, ∞[; due to the monotonicity Properties 2 (P10) to (P12) of the sequences *a* (*q L λ* ) *n n*∈N and *b* (*p L λ* ,*q L λ* ) *n n*∈N , these bounds have led to the "optimal" recursive lower bound *B L λ*,*X*0,*n* of the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*) in (40) of Theorem 1(b)).

**Figure 3.** The lower bound E *L n* (using *λ* = 1.1) and the upper bound E *U n* (using *λ* = 0.5) of the minimal type II error probability E*<sup>ς</sup>* (*P*A,*n*||*P*H,*n*) for different levels *ς* ∈ {0.01, 0.05, 0.1} in the parameter setup (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) = (0.3, 1.2, 1, 4) ∈ PSP,1 together with initial population *<sup>X</sup>*<sup>0</sup> = 5 on both a unit scale (**left graph**) and a logarithmic scale (**right graph**).

In contrast, the strict *convexity* of the function *φλ*(·) in the case (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (PSP\PSP,1) × (R\[0, 1]) implies that we cannot maximize both parameters *p L λ* , *q L <sup>λ</sup>* ∈ R *simultaneously* subject to the constraint (35). This effect carries over to the lower bounds *B L λ*,*X*0,*n* of the Hellinger integrals *Hλ*(*P*A,*n*||*P*H,*n*) (cf. (41)); in general, these bounds cannot be maximized *simultaneously* for all initial population sizes *X*<sup>0</sup> ∈ N and all observation horizons *n* ∈ N.

Analogously to (46), one way to obtain "good" recursive lower bounds for *Hλ*(*P*A,*n*||*P*H,*n*) from (41) in Theorem 1 (b) is to solve the optimization problem,

$$\mathbb{E}\left\{\overline{p\_{\lambda'}^{L}}\overline{q\_{\lambda}^{L}}\right\} := \underset{(p\_{\lambda}^{L},q\_{\lambda}^{L})\in\mathbb{R}^{2}}{\arg\max} \left\{ \exp\left\{a\_{n}^{(q\_{\lambda}^{L})} \cdot \mathbf{X}\_{0} + \sum\_{k=1}^{n} b\_{k}^{(p\_{\lambda}^{L},q\_{\lambda}^{L})}\right\} \right\} \qquad \text{such that (35) is satisfied,} \tag{55}$$

for each fixed initial population size *X*<sup>0</sup> ∈ N and observation horizon *n* ∈ N. But due to the same reasons as explained right after (46), the optimization problem (55) seems to be not straightforward to solve explicitly. In a congeneric way as in the discussion of the upper bounds for the case *λ* ∈]0, 1[ above, we now have to look for suitable parameters *p L λ* , *q L λ* for the lower bound *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≤ *Hλ*(*P*A,*n*||*P*H,*n*) that fulfill (35) and that guarantee certain reasonable criteria and goals; these are similar to the goals (G1) to (G3) from Section 3.6, and are therefore supplemented by an additional " 0 ":

(G10 ) the validity of *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* > 1 *simultaneously* for all initial configurations *X*<sup>0</sup> ∈ N, all observation horizons *n* ∈ N and all *λ* ∈ R\[0, 1], which leads to a *strict* improvement of the general upper bound *Hλ*(*P*A,*n*||*P*H,*n*) > 1 (cf. (11));


In the following, let us briefly discuss how these three goals can be achieved in principle, where we confine ourselves to parameters *p L λ* , *q L <sup>λ</sup>* which–in addition to (35)–fulfill the requirement

$$\left\{ \begin{array}{ll} q\_{\lambda}^{L} \geq \max\{0, \mathfrak{f}\_{\lambda}\} & \wedge & p\_{\lambda}^{L} > \max\{0, a\_{\lambda}\} \end{array} \right\} \quad \vee \quad \left\{ \begin{array}{ll} q\_{\lambda}^{L} > \max\{0, \mathfrak{f}\_{\lambda}\} & \wedge & p\_{\lambda}^{L} \geq \max\{0, a\_{\lambda}\} \end{array} \right\},\tag{56}$$

where ∧ is the logical "AND" and ∨ the logical "OR" operator. This is sufficient to tackle all three Goals (G10 ) to (G30 ). To see this, assume that *p L λ* , *q L λ* satisfy (35). Let us begin with the two "extremal" cases in (56), i.e., with (i) *q L <sup>λ</sup>* = max{0, *βλ*}, *p L <sup>λ</sup>* > max{0, *αλ*}, respectively (ii) *q L <sup>λ</sup>* > max{0, *βλ*}, *p L <sup>λ</sup>* = max{0, *αλ*}.

Suppose in the first extremal case (i) that *β<sup>λ</sup>* ≤ 0. Then, *q L <sup>λ</sup>* = 0 and Properties 1 (P4) implies that *a* (*q L λ* ) *<sup>n</sup>* = −*β<sup>λ</sup>* ≥ 0 and hence *b* (*p L λ* ,*q L λ* ) *<sup>n</sup>* = *p L λ e* <sup>−</sup>*β<sup>λ</sup>* <sup>−</sup> *<sup>α</sup><sup>λ</sup>* <sup>≥</sup> *<sup>p</sup> L <sup>λ</sup>* − *α<sup>λ</sup>* > 0 for all *n* ∈ N. This enters into (41) as follows: the Hellinger integral lower bound becomes *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≥ *B*e (*p L λ* ,*q L λ* ) *<sup>λ</sup>*,*X*0,*<sup>n</sup>* = exp{−*β<sup>λ</sup>* · *X*<sup>0</sup> + (*p L λ e* <sup>−</sup>*β<sup>λ</sup>* <sup>−</sup> *αλ*) · *n*} > 1. Furthermore, one clearly has lim*n*→<sup>∞</sup> *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = ∞ as well as lim*n*→<sup>∞</sup> 1 *n* log *B L λ*,*X*0,*n* = *p L λ e* <sup>−</sup>*β<sup>λ</sup>* <sup>−</sup> *<sup>α</sup><sup>λ</sup>* <sup>&</sup>gt; 0. Assume now that *<sup>β</sup><sup>λ</sup>* <sup>&</sup>gt; 0. Then, *<sup>q</sup> L <sup>λ</sup>* = *β<sup>λ</sup>* > 0, *a* (*q L λ* ) *<sup>n</sup>* = 0 (cf. (P2)), *b* (*p L λ* ,*q L λ* ) *<sup>n</sup>* = *p L <sup>λ</sup>* − *α<sup>λ</sup>* > 0 and thus *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = exp{(*p L <sup>λ</sup>* − *αλ*) · *n*} > 1 for all *n* ∈ N. Furthermore, one gets lim*n*→<sup>∞</sup> *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = ∞ as well as lim*n*→<sup>∞</sup> 1 *n* log *B L λ*,*X*0,*n* = *p L <sup>λ</sup>* − *α<sup>λ</sup>* > 0.

Let us consider the other above-mentioned extremal case (ii). Suppose that *q L <sup>λ</sup>* > max{0, *βλ*} together with *q L <sup>λ</sup>* > min{1,*e <sup>β</sup>λ*−1} which implies that the sequence *a* (*q L λ* ) *n n*∈N is strictly positive, strictly increasing and grows to infinity faster than exponentially, cf. (P3b). Hence, *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≥ exp{*a* (*q L λ* ) *<sup>n</sup>* · *X*0} > 1, lim*n*→<sup>∞</sup> *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = ∞ as well as lim*n*→<sup>∞</sup> 1 *n* log *B L λ*,*X*0,*n* = ∞. If max{0, *βλ*} < *q L <sup>λ</sup>* ≤ min{1,*e <sup>β</sup>λ*−1}, then *a* (*q L λ* ) *n n*∈N is strictly positive, strictly increasing and converges to *x* (*qλ*) <sup>0</sup> ∈]0, − log(*q L λ* )] (cf. (P3a)). This carries over to the sequence *b* (*p L λ* ,*q L λ* ) *n n*∈N : one gets *b* (*p L λ* ,*q L λ* ) <sup>1</sup> = *p L <sup>λ</sup>* − *α<sup>λ</sup>* ≥ 0 and *b* (*p L λ* ,*q L λ* ) *<sup>n</sup>* > 0 for all *n* ≥ 2. Furthermore, *b* (*p L λ* ,*q L λ* ) *<sup>n</sup>* is strictly increasing and converges to *p L λ* · *e x* (*q L λ* ) <sup>0</sup> − *α<sup>λ</sup>* > 0, leading to *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* > 1 for all *n* ∈ N, to lim*n*→<sup>∞</sup> *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = ∞ as well as to lim*n*→<sup>∞</sup> 1 *n* log *B L λ*,*X*0,*n* = *p L λ* · *e x* (*q L λ* ) <sup>0</sup> − *α<sup>λ</sup>* > 0.

It remains to look at the cases where *p L λ* , *q L λ* satisfy (35), and (56) with two strict inequalities. For this situation, one gets

• *a* (*q L λ* ) *n n*∈N is strictly positive, strictly increasing and–iff *q L <sup>λ</sup>* ≤ min{1,*e <sup>β</sup>λ*−1}–convergent namely to the smallest positive solution *x* (*q L λ* ) <sup>0</sup> ∈]0, − log(*q L* )] of (44) , cf. (P3);

*λ*

• *b* (*p L λ* ,*q L λ* ) *n n*∈N is strictly increasing, strictly positive since *b* (*p L λ* ,*q L λ* ) <sup>1</sup> = *p L <sup>λ</sup>* − *α<sup>λ</sup>* > 0 and–iff *q L λ* ≤ min{1,*e <sup>β</sup>λ*−1}–convergent namely to *p L λ e x* (*q L λ* ) <sup>0</sup> −*α<sup>λ</sup>* ∈ [*p L <sup>λ</sup>* − *αλ*, *p L λ* /*q L <sup>λ</sup>* − *αλ*] , cf (P7).

Hence, under the assumptions (35) and *p L <sup>λ</sup>* > max{0, *αλ*} ∧ *q L <sup>λ</sup>* > max{0, *βλ*} the corresponding lower bounds *B L λ*,*X*0,*n* of the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*) fulfill for all *X*<sup>0</sup> ∈ N

• *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* > 1 for all *n* ∈ N,


Putting these considerations together we conclude that the constraints (35) and (56) are sufficient to achieve the Goals (G10 ) to (G30 ). Hence, for fixed parameter constellation (*β*A, *β*H, *α*A, *α*H, *λ*), we aim for finding *p L <sup>λ</sup>* = *p L* (*β*A, *β*H, *α*A, *α*H, *λ*) and *q L <sup>λ</sup>* = *q L* (*β*A, *β*H, *α*A, *α*H, *λ*) which satisfy (35) and (56). This can be achieved mostly, but not always, as we shall show below. As an auxiliary step for further investigations, it is useful to examine the set of all *λ* ∈ R\[0, 1] for which *α<sup>λ</sup>* ≤ 0 or *β<sup>λ</sup>* ≤ 0 (or both). By straightforward calculations, we see that

$$a\_{\lambda} \le 0 \iff \lambda \left\{ \begin{array}{ll} \le & \frac{-a\_{\mathcal{H}}}{\overline{a}\_{\mathcal{A}} - a\_{\mathcal{H}}}, \text{ if } a\_{\mathcal{A}} > a\_{\mathcal{H}}, \\\\ \ge & \frac{a\_{\mathcal{H}}}{\overline{a}\_{\mathcal{H}} - a\_{\mathcal{A}'}}, \text{ if } a\_{\mathcal{A}} < a\_{\mathcal{H}}, \end{array} \right. \\ \left. \begin{array}{ll} \text{and} & \begin{array}{ll} \left. \le & \frac{-\beta\_{\mathcal{H}}}{\overline{\beta\_{\mathcal{A}} - \overline{\beta\_{\mathcal{H}}}}}, \text{ if } \beta\_{\mathcal{A}} > \beta\_{\mathcal{H}}, \\\\ \ge & \frac{\beta\_{\mathcal{H}}}{\overline{\beta\_{\mathcal{H}} - \overline{\beta\_{\mathcal{A}}}}}, \text{ if } \beta\_{\mathcal{A}} < \beta\_{\mathcal{H}}. \end{array} \right. \end{array} \right. \right\} \tag{57}$$

Furthermore, recall that (35) implies the general bounds *p L <sup>λ</sup>* ≤ *α λ* A *α* 1−*λ* <sup>H</sup> <sup>=</sup> *<sup>ϕ</sup>λ*(0) (being equivalent to the requirement *φ L λ* (0) = *φλ*(0) ) and *q L <sup>λ</sup>* ≤ *β λ* A *β* 1−*λ* <sup>H</sup> <sup>=</sup> *<sup>q</sup>*e*<sup>λ</sup>* (the latter being the maximal slope due to Properties 3 (P19), (P20)).

Let us now undertake the desired *detailed* investigations on lower and upper bounds of the Hellinger integrals *Hλ*(*P*A,*n*||*P*H,*n*) of order *λ* ∈ R\[0, 1], for the various different subclasses of PSP\PSP,1.
