Berry-Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment

Abstract

Let $(Z_n)$ be a supercritical branching process in a random environment $\xi =({\xi }_n)$. We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for $\log Z_n$ under the annealed law $\mathbb{P}$. We also improve some earlier results about the harmonic moments of the limit variable $W=li{m}_{n\to \infty }{W}_n$, where $W_n=Z_n/{\mathbb{E}}_{\xi }{Z}_n$ is the normalized population size.

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• Asymptotic of the distribution and harmonic moments for a supercritical branching process in a random environment