Exact convergence rate of the local limit theorem for branching random walks on the integer lattice

Consider branching random walks on the integer lattice Zd, where the branching mechanism is governed by a supercritical Galton-Watson process and the particles perform a symmetric nearest-neighbor random walk whose increments equal to zero with probability r∈[0,1). We derive exact convergence rate in the local limit theorem for distributions of particles. When r=0, our results correct and improve the existing results on the convergence speed conjectured by Révész (1994) and proved by Chen (2001). As a byproduct, we obtain exact convergence rate in the local limit theorem for some symmetric nearest-neighbor random walks, which is of independent interest.