Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

TL;DR

This paper derives second and third order asymptotic expansions for the distribution of particles in a branching random walk with a random environment, enhancing understanding of higher order effects in such stochastic processes.

Contribution

It introduces new higher order asymptotic expansions for the distribution of particles in a branching random walk with a random environment, using novel martingale techniques.

Findings

Second and third order asymptotic expansions obtained

New martingale with proven convergence rate introduced

Higher order effects in branching random walks elucidated

Abstract

Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time.For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.