Exponential and laplace approximation for occupation statistics of branching random walk

Abstract

We study occupancy counts for the critical nearest-neighbor branching random walk on the d-dimensional lattice, conditioned on non-extinction. For d 3, Lalley and Zheng [4] showed that the properly scaled joint distribution of the number of sites occupied by j generation-n particles, j = 1, 2, …, converges in distribution as n goes to infinity, to a deterministic multiple of a single exponential random variable. The limiting exponential variable can be understood as the classical Yaglom limit of the total population size of generation n. Here we study the second order fluctuations around this limit, first, by providing a rate of convergence in the Wasserstein metric that holds for all d 3, a..