Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

TL;DR

This paper extends and sharpens existing results on convergence rates in central limit theorems for branching random walks, under weaker moment conditions and in a more general random environment in time.

Contribution

It generalizes Chen's results by relaxing moment conditions and incorporating a random environment, introducing new terms in rate functions and handling technical challenges with martingale analysis.

Findings

Derived exact convergence rates under weaker moment conditions.

Identified new terms in rate functions for branching random walks.

Established results in a more general framework with random environments.

Abstract

Chen [Ann. Appl. Probab. {\bf 11} (2001), 1242--1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process.We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form $ \E X (\ln^+X )^{1+\lambda}< \infty$. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work.The results are established in the more general framework, i.e. for a branching random walk with a random environment in time.The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment.