Harmonic moments and large deviations for a critical Galton-Watson process with immigration

In this paper,a critical Galton-Watson branching process with immigration Z;is studied.We first obtain the convergence rate of the harmonic moment of Z;.Then the large deviation of S;:∑;X;is obtained,where {X;} is a sequence of independent and identically distributed zero-mean random variables with the tail index α> 2.We shall see that the converging rate is determined by the immigration mean,the variance of reproducing and the tail index of X;,compared with the previous result for the supercritical case,where the rate depends on the Schroder constant and the tail index.