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

TL;DR

This paper establishes large and moderate deviation principles, harmonic moment criteria, and central limit theorems for a supercritical branching process in a random environment, enhancing understanding of its probabilistic behavior.

Contribution

It introduces new large and moderate deviation results and harmonic moment conditions for branching processes in random environments, along with associated central limit theorems.

Findings

Large and moderate deviation principles for log Z_n

Critical harmonic moment value for W

Central limit theorems for W-W_n and log Z_n

Abstract

Let $(Z_{n})$ be a supercritical branching process in a random environment $\xi $, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|\xi ]$. 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.