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 $Z_{n}/\mathbb{E}[Z_{n}|ξ]$. 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.

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