LAUSR

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 Revesz (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.