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

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