Harmonic moments and large deviation rates for supercritical branching processes

Let {Zn,n.1} be a single type supercritical Galton--Watson process with mean EZ1.m, initiated by a single ancestor. This paper studies the large deviation behavior of the sequence {Rn.Zn+1Zn\dvtxn.1} and establishes a "phase transition" in rates depending on whether r, the maximal number of moments possessed by the offspring distribution, is less than, equal to or greater than the Schröder constant .. This is done via a careful analysis of the harmonic moments of Zn.