Times of a Branching Process with Immigration in Varying Environments Attaining a Fixed Level

Consider a branching process {Z_n}_n≥0 with immigration in varying environments. For a ∈ {0, 1, 2, …}, let C(a) = {n ≥ 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(a) is finite or not. For the critical Galton–Watson process, based on a moment method, we show that \({{| {C(a) \cap [1,n]} |} \over {\log \;n \to S}}\) in distribution, where S is an exponentially distributed random variable with P(S > t) = e^−t, t > 0.