Exact convergence rate in the local central limit theorem for a lattice branching random walk on [formula omitted

Abstract

Consider a branching random walk, where the branching mechanism is governed by a Galton–Watson process and the migration is governed by a lattice random walk on Zd. Under the mild moment conditions for the underlying branching mechanism and migration laws, we figure out the exact convergence rate of the local limit theorem for the counting measure which counts the number of particles of generation n in a given set. Our result extends the previous one obtained by Chen (2001) for a simple branching random walk with second moment condition for the offspring law.