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

TL;DR

This paper develops probabilistic bounds and large deviation estimates for the logarithm of a supercritical branching process in a random environment, enhancing understanding of its probabilistic behavior.

Contribution

It introduces new Berry-Esseen bounds and Cramér-type large deviation expansions for the process, and improves existing results on harmonic moments of the limiting variable.

Findings

Established a Berry-Esseen bound for log Z_n.

Derived a Cramér's large deviation expansion for log Z_n.

Improved results on harmonic moments of the limit variable W.

Abstract

Let $(Z_n)$ be a supercritical branching process in a random environment $\xi = (\xi_n)$. We establish a Berry-Esseen bound and a Cram\'er'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 $W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_{\xi} Z_n$ is the normalized population size.