Harmonic moments and large deviations for a critical Galton-Watson process with immigration. (English) Zbl 1476.60156
Summary: 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 \({S_{{Z_n}}}:= \sum\nolimits_{i = 1}^{{Z_n}} {{X_i}}\) is obtained, where \(\{X_i\}\) is a sequence of independent and identically distributed zero-mean random variables with the 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 \(X_1^+\), compared with the previous result for the supercritical case, where the rate depends on the Schröder constant and the tail index.
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