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 ≥ 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≤i≤n Z_i is obtained.