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 $\xi$, and $W$ be the limit of the normalized population size $Z_n\mathrm{/}\mathbb{E}[Z_n\mathrm{\mid }\xi ]$. 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.