Exponential and Laplace approximation for occupation statistics of branching random walk

TL;DR

This paper investigates the detailed probabilistic behavior of occupancy counts in critical branching random walks on lattices, establishing convergence rates and identifying Laplace distribution limits for fluctuations in high dimensions.

Contribution

It provides the first rates of convergence for occupancy statistics and demonstrates Laplace distribution limits for fluctuations in dimensions $d extgreater=7$, extending previous results.

Findings

Convergence rates in Wasserstein metric for occupancy counts.

Laplace distribution as the limit for scaled fluctuations in high dimensions.

Extension of previous results to all $d extgreater=3$ for convergence rates.

Abstract

We study occupancy counts for the critical nearest-neighbor branching random walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq 3$, Lalley and Zheng (2011) showed that the properly scaled joint distribution of the number of sites occupied by $j$ generation-$n$ particles, $j=1,2,\ldots$, 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 $d\geq3$, and second, by showing that for $d\geq 7$, the weak limit of the scaled joint differences between the number of occupancy-$j$ sites and appropriate multiples of the total population size converge in the Wasserstein metric to a multivariate symmetric Laplace distribution. We also provide a rate of convergence for this latter result.