Limit theorems for a critical branching process with immigration at zero in a random environment

Abstract

In this paper, we study a critical branching process with immigration at zero in a random environment $Z$, where the immigration is allowed entering a generation iff the previous generation is empty. Firstly, we analyze the asymptotic behavior of the survival probability at generation $n$ of $\tilde{Z}$, a critical branching process in a random environment with random initial number of particles. Then, the convergence rate of the transition probabilities $p_{k,0}^{\left(n\right)}$ ($k\ge 0$) of $Z$ are obtained, which can be compared with the corresponding results for non-random case in literatures.