Limit theorems for supercritical branching processes in random environment

TL;DR

This paper develops Fourier-based techniques to analyze supercritical branching processes in random environments, providing new limit theorems such as CLT, Edgeworth expansions, and renewal results for the logarithm of the process size.

Contribution

It introduces Fourier methods to derive precise asymptotic results for supercritical branching processes in random environments, advancing theoretical understanding.

Findings

Established a central limit theorem for log Z_n

Derived Edgeworth expansions for the process

Proved renewal theorems related to the process growth

Abstract

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $\log Z_n$.