Harmonic Moments for Supercritical Multi-type Galton–Watson Processes

In this paper, the convergence rates of harmonic moment E [( 1 · Z n ) − r ] of a supercritical multi-type Galton–Watson process { Z n ; n ≥ 0} are studied. It is shown that there exists a phase transition in the convergence rate of the harmonic moments, which extends the results from single type cases to multi-type cases. Based on above result, under some moment hypothesis on Z n , the large deviations of l ⋅ Z n + 1 1 ⋅ Z n are studied for l > 0 . In particular, in the case that the offspring distributions are of Pareto-type, an explicit large deviation result is obtained.