Exact convergence rate of the local limit theorem for a branching random walk in a time-dependent random environment on d-dimensional integer lattice

Abstract

Consider a branching random walk in where both the offspring distribution and the displacement law vary with generation time. For each let denote the number of particles of n-th generation located at x. We derive exact convergence rate of the local limit theorem for the counting measure This generalizes the result obtained in Gao (2017, SPA) by adding the random environment affects, and improves it by weakening the moment condition required for the offspring distribution.