Regeneration of branching processes with immigration in varying environments

TL;DR

This paper studies linear-fractional branching processes with immigration in changing environments, providing criteria for regeneration time behavior and illustrating complex phenomena like extinction with finitely many regeneration times.

Contribution

It offers new criteria for the finiteness of regeneration times and constructs examples demonstrating unique behaviors caused by varying environments.

Findings

Finiteness or infiniteness criteria for regeneration times.

Existence of processes extinct with finitely many regeneration times.

Bound on the number of regeneration times in large intervals.

Abstract

In this paper, we consider certain linear-fractional branching processes with immigration in varying environments. For $n\ge0,$ let $Z_n$ counts the number of individuals of the $n$-th generation, which excludes the immigrant which enters into the system at time $n.$ We call $n$ a regeneration time if $Z_n=0.$ We give first a criterion for the finiteness or infiniteness of the number of regeneration times. Then, we construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. It may happen that the process is extinct but there are only finitely many regeneration times. Also, when there are infinitely many regeneration times, we show that for each $\varepsilon>0,$ the number of regeneration times in $[0,n]$ is no more than $(\log n)^{1+\varepsilon}$ as $n\rightarrow\infty.$