Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in ℤ^d with a Random Environment in Time

Consider a branching random walk with a random environment in time in the d-dimensional integer lattice. The branching mechanism is governed by a supercritical branching process, and the particles perform a lazy random walk with an independent, non-identical increment distribution. For A ⊂ ℤ^d, let ℤ_n(A) be the number of offsprings of generation n located in A. The exact convergence rate of the local limit theorem for the counting measure Z_n(·) is obtained. This partially extends the previous results for a simple branching random walk derived by Gao (2017, Stoch. Process Appl.).