Central limit theorem and Berry-Esseen bounds for a branching random walk with immigration in a random environment

TL;DR

This paper establishes a central limit theorem and Berry-Esseen bounds for a branching random walk with immigration in a random environment, providing precise convergence rates and insights into the process's dispersion over time.

Contribution

It introduces new Berry-Esseen bounds and convergence rates for the branching random walk with immigration in a random environment, extending existing CLT results.

Findings

Central limit theorem holds for the partition function logarithm.

Established uniform and non-uniform Berry-Esseen bounds.

Discovered the exact convergence rate in the CLT.

Abstract

We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\mathbf t$ fixed, the logarithm $\log Z_n(\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the central limit theorem, and discover the exact convergence rate in the central limit theorem.