Asymptotic Behaviors for Critical Branching Processes with Immigration

In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\{Z_n, n\ge 0\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\max_{1\le i\le n} Z_i$ is obtained.