A Second Order Expansion in the Local Limit Theorem for a Branching System of Symmetric Irreducible Random Walks

Z.-Q. Gao

2021, v.27, Issue 3, 439-466

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 $\mathbb{Z}^d$.
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Let $Z_n(z)$ be the number of the particles in the $n$-th generation at the point $z\in \Z^d$.
Under the mild moment conditions for offspring distribution of the underlying Galton\tire Watson, we derive a second order expansion in the local limit theorem for $Z_n(z)$ for each given $z\in \Z^d$. That generalises the results for simple branching random walks obtained by Gao \cite{Gao2018SPA}.
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%For the case where the underlying moving mechanism is governed by simple random walks, this kind of expansion was obtained by the author in 2018.

Keywords: branching random walk, local limit theorem, second order expansion

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