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

Abstract

Consider a branching process $\{Z_n{\}}_{n\ge 0}$ with immigration in varying environment. For $a\in \{0,1,2,...\},$ let $C=\{n\ge 0:Z_n=a\}$ 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 $|C\cap [1,n]|/\log n\to S$ in distribution, where $S$ is an exponentially distributed random variable with $P(S>t)=e^{-t},t>0.$

20 pages