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 $C={n\ge0:Z_n=a}$ 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 $P(S>t)=e^{-t},\ t>0.$