Almost sure central limit theorem for branching random walks in random environment

TL;DR

This paper establishes an almost sure central limit theorem for the density of branching random walks in random environments on high-dimensional lattices, under certain integrability conditions, revealing detailed population fluctuation behavior.

Contribution

It proves a new almost sure central limit theorem for branching random walks in random environments, extending understanding of population density fluctuations in high dimensions.

Findings

The normalized total population converges almost surely to a nondegenerate random variable.

A central limit theorem is established for the population density in high dimensions.

The results depend on uniform square integrability of environmental fluctuations.

Abstract

We consider the branching random walks in $d$-dimensional integer lattice with time--space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable. When $d\geq3$ and the fluctuation of environment satisfies a certain uniform square integrability then it is nondegenerate and we prove a central limit theorem for the density of the population in terms of almost sure convergence.