Times of a branching process with immigration in varying environment attaining a fixed level

TL;DR

This paper investigates the times at which a branching process with immigration in a changing environment reaches a fixed population level, providing criteria for finiteness and asymptotic distribution results.

Contribution

It introduces a criterion to determine the finiteness of times the process hits a fixed level and characterizes the asymptotic distribution for critical processes.

Findings

Set C is finite or infinite based on the environment.

For critical processes, the scaled count of hitting times converges to an exponential distribution.

Provides a new understanding of hitting times in varying environments.

Abstract

Consider a branching process $\{Z_n\}_{n\ge 0}$ with immigration in varying environment. For $a\in\{0,1,2,...\},$ let $C=\{n\ge0:Z_n=a\}$ be the collection of times at which the population size of the process attains level $a.$ We give a criterion to determine whether the set $C$ is finite or not. For critical Galton-Watson process, we show that $|C\cap [1,n]|/\log n\rightarrow S$ in distribution, where $S$ is an exponentially distributed random variable with $P(S>t)=e^{-t},\ t>0.$