Exact convergence rate in the central limit theorem for a branching process with immigration in a random environment

TL;DR

This paper establishes the precise rate at which the distribution of the logarithm of a branching process with immigration in a random environment converges to a normal distribution, using advanced probabilistic techniques.

Contribution

It provides the exact convergence rate in the CLT for the process's logarithm, extending understanding of convergence behavior in complex stochastic systems.

Findings

Derived the exact convergence rate in the CLT for logZn

Connected convergence rates of submartingales to the process

Applied Berry-Esseen bounds to the random walk component

Abstract

Let (Zn) be a branching process with immigration in an independent and identically distributed random environment. Under necessary moment conditions, we show the exact convergence rate in the central limit theorem on logZn by using the convergence rates of the logarithm of submartingale and the result of the corresponding random walk on the Berry-Esseen bound.