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Consider a branching random walk, where the branching mechanism is governedby a Galton-Watson process, and the migration by a finite range symmetricirreducible random walk on the integer lattice $\mathbb{Z}^d$. Let $Z_n(z)$ bethe 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 ofthe underlying Galton-Watson, we derive a second order expansion in the locallimit theorem for $Z_n(z)$ for each given $z\in \mathbb{Z}^d$. That generalizesthe results for simple branching random walks obtained by Gao [2018, SPA].

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