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

Abstract

In this paper, a critical Galton-Watson branching process with immigration $Z_n$ is studied. We first obtain the convergence rate of the harmonic moment of $Z_n$. Then the large deviation of $S_{Z_n}:={\sum }_{i=1}^{Z_n}{X}_i$ is obtained, where $\{X_i\}$ is a sequence of independent and identically distributed zero-mean random variables with tail index $\alpha >2$. We shall see that the converging rate is determined by the immigration mean, the variance of reproducing and the tail index of $X_1^{+}$, comparing to previous result for supercritical case, where the rate depends on the Schröder constant and the tail index.

17 pages