Some distance bounds of branching processes and their diffusion limits

TL;DR

This paper derives exact bounds on distances between Poisson-distributed Galton-Watson branching processes with immigration and explores their diffusion limits, with implications for statistical distinguishability and decision-making.

Contribution

It provides novel bounds on divergences between general GWI processes and characterizes their diffusion limits, extending understanding of their asymptotic behavior.

Findings

Exact bounds for divergences between GWI processes.

Asymptotic behavior of GWI under diffusion limits.

Applications to Bayesian decision making and hypothesis testing.

Abstract

We compute exact values respectively bounds of "distances" - in the sense of (transforms of) power divergences and relative entropy - between two discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration is arbitrarily Poisson-distributed (leading to arbitrary type of criticality). Implications for asymptotic distinguishability behaviour in terms of contiguity and entire separation of the involved GWI are given, too. Furthermore, we determine the corresponding limit quantities for the context in which the two GWI converge to Feller-type branching diffusion processes, as the time-lags between observations tend to zero. Some applications to (static random environment like) Bayesian decision making and Neyman-Pearson testing are presented as well.