Limit Behaviors for a Critical Galton-Watson Process with Immigration

We consider a critical Galton-Watson branching process with immigration Z_n, and study the convergence rate of the harmonic moments of this process, improving the results in previous literatures. The proof is based on the local probabilities estimations of Z_n. As applications, we obtain the large deviations of S_Z_n:=\tsm_i=1^Z_nX_i, where \X_i,i\geq 1\ is a sequence of independent and identically distributed random variables, and X_1 is in the domain of attraction of an \alpha-stable law with \alpha\in(0,2).