Moments, moderate and large deviations for a branching process in a random environment

Let (Zn)(Zn) be a supercritical branching process in a random environment ξξ, and WW be the limit of the normalized population size Zn/E[Zn|ξ]Zn/E[Zn|ξ]. We show large and moderate deviation principles for the sequence logZnlogZn (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of WW, and show an equivalence for all the moments of ZnZn. Central limit theorems on W−WnW−Wn and logZnlogZn are also established.