A second order expansion in the local limit theorem for a branching system of symmetric irreducible random walks

TL;DR

This paper derives a second order expansion in the local limit theorem for a branching random walk with symmetric irreducible random walks, generalizing previous results for simple branching random walks.

Contribution

It introduces a second order expansion in the local limit theorem for branching random walks with finite range symmetric jumps, extending Gao's 2018 results.

Findings

Second order expansion in local limit theorem derived

Generalizes results from simple to more complex branching random walks

Applicable under mild moment conditions for offspring distribution

Abstract

Consider a branching random walk, where the branching mechanism is governed by a Galton-Watson process, and the migration by a finite range symmetric irreducible random walk on the integer lattice $\mathbb{Z}^d$. Let $Z_n(z)$ be the number of the particles in the $n$-th generation at the point $z\in \mathbb{Z}^d$. Under the mild moment conditions for offspring distribution of the underlying Galton-Watson, we derive a second order expansion in the local limit theorem for $Z_n(z)$ for each given $z\in \mathbb{Z}^d$. That generalizes the results for simple branching random walks obtained by Gao [2018, SPA].