1 Introduction	3
2 The Framework and Application Setups	5
2.1 Process Setup	5
2.2 Connections to Time Series of Counts	6
2.3 Applicability to Epidemiology	8
2.4 Information Measures	12
2.5 Decision Making under Uncertainty	15
2.6 Asymptotical Distinguishability	19
3 Detailed Recursive Analyses of Hellinger Integrals	21
3.1 A First Basic Result	21
3.2 Some Useful Facts for Deeper Analyses	25
3.3 Detailed Analyses of the Exact Recursive Values, i.e., for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}}\right)\in {\mathcal{P}}_{\mathrm{NI}}\cup {\mathcal{P}}_{\mathrm{SP},1}$	27
3.4 Some Preparatory Basic Facts for the Remaining Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}}\right)\in {\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1}$	29
3.5 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times ]0,1[$	31
3.6 Goals for Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times ]0,1[$	32
3.7 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},2}\times ]0,1[$	34
3.8 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},3\mathrm{a}}\times ]0,1[$	35
3.9 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},3\mathrm{b}}\times ]0,1[$	36
3.10 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},3\mathrm{c}}\times ]0,1[$	37
3.11 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},4\mathrm{a}}\times ]0,1[$	37
3.12 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},4\mathrm{b}}\times ]0,1[$	37
3.13 Concluding Remarks on Alternative Upper Bounds for all Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in$ $({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times ]0,1[$	37
3.14 Intermezzo 1: Application to Asymptotical Distinguishability	38
3.15 Intermezzo 2: Application to Decision Making under Uncertainty	39
3.15.1 Bayesian Decision Making	39
3.15.2. Neyman-Pearson Testing	41
3.16 Goals for Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash [0,1])$	41
3.17 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},2}\times (\mathbb{R}\backslash [0,1])$	44
3.18 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},3\mathrm{a}}\times (\mathbb{R}\backslash [0,1])$	45
3.19 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},3\mathrm{b}}\times (\mathbb{R}\backslash [0,1])$	46
3.20 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},3\mathrm{c}}\times (\mathbb{R}\backslash [0,1])$	47
3.21 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},4\mathrm{a}}\times (\mathbb{R}\backslash [0,1])$	47
3.22 Lower Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in {\mathcal{P}}_{\mathrm{SP},4\mathrm{b}}\times (\mathbb{R}\backslash [0,1])$	48
3.23 Concluding Remarks on Alternative Lower Bounds for all Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash [0,1])$	48
3.24 Upper Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash [0,1])$	48
4 Power Divergences of Non-Kullback-Leibler-Information-Divergence Type	49
4.1 A First Basic Result	49
4.2 Detailed Analyses of the Exact Recursive Values of $I_{\lambda }(·\parallel ·)$, i.e., for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{NI}}\cup {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash \{0,1\})$	51
4.3 Lower Bounds of $I_{\lambda }(·\parallel ·)$ for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times ]0,1[$	52
4.4 Upper Bounds of $I_{\lambda }(·\parallel ·)$ for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times ]0,1[$	53
4.5 Lower Bounds of $I_{\lambda }(·\parallel ·)$ for the Cases $({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda )\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash [0,1])$	53
4.6 Upper Bounds of $I_{\lambda }(·\parallel ·)$ for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash [0,1])$	54
4.7 Applications to Bayesian Decision Making	55
5 Kullback-Leibler Information Divergence (Relative Entropy)	55
5.1 Exact Values Respectively Upper Bounds of $I(·||·)$	55
5.2 Lower Bounds of $I(·||·)$ for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}}\right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})$	56
5.3 Applications to Bayesian Decision Making	58
6 Explicit Closed-Form Bounds of Hellinger Integrals	59
6.1 Principal Approach	59
6.2 Explicit Closed-Form Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{NI}}\cup {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash \{0,1\})$	63
6.3 Explicit Closed-Form Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times ]0,1[$	64
6.4 Explicit Closed-Form Bounds for the Cases $\left({\beta }_{\mathcal{A}},{\beta }_{\mathcal{H}},{\alpha }_{\mathcal{A}},{\alpha }_{\mathcal{H}},\lambda \right)\in ({\mathcal{P}}_{\mathrm{SP}}\backslash {\mathcal{P}}_{\mathrm{SP},1})\times (\mathbb{R}\backslash [0,1])$	67
6.5 Totally Explicit Closed-Form Bounds	69
6.6 Closed-Form Bounds for Power Divergences of Non-Kullback-Leibler-Information-Divergence Type	70
6.7 Applications to Decision Making	71
7 Hellinger Integrals and Power Divergences of Galton-Watson Type Diffusion Approximations	71
7.1 Branching-Type Diffusion Approximations	71
7.2 Bounds of Hellinger Integrals for Diffusion Approximations	74
7.3 Bounds of Power Divergences for Diffusion Approximations	79
7.4 Applications to Decision Making	80
A Proofs and Auxiliary Lemmas	81
A.1. Proofs and Auxiliary Lemmas for Section 3	81
A.2 Proofs and Auxiliary Lemmas for Section 5	88
A.3 Proofs and Auxiliary Lemmas for Section 6	94
A.4 Proofs and Auxiliary Lemmas for Section 7	101
References	115