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 Zn is studied. We first obtain the convergence rate of the harmonic moment of Zn. Then the large deviation of $${S_{{Z_n}}}≔ \sum\nolimits_{i = 1}^{{Z_n}} {{X_i}} $$ is obtained, where {Xi} 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 1 + , compared with the previous result for the supercritical case, where the rate depends on the Schroder constant and the tail index.