Wasserstein-$1$ distance and nonuniform Berry-Esseen bound for a supercritical branching process in a random environment

TL;DR

This paper establishes optimal convergence rates in Wasserstein-1 distance and exponential Berry-Esseen bounds for supercritical branching processes in random environments, with applications to confidence interval estimation.

Contribution

It provides the first optimal Wasserstein-1 convergence rate and exponential Berry-Esseen bounds for such processes, extending previous results.

Findings

Optimal Wasserstein-1 convergence rate established

Exponential nonuniform Berry-Esseen bounds derived

Applications to confidence interval estimation for criticality and population size

Abstract

Let $ (Z_{n})_{n\geq 0} $ be a supercritical branching process in an independent and identically distributed random environment. We establish an optimal convergence rate in the Wasserstein-$1$ distance for the process $ (Z_{n})_{n\geq 0} $, which completes a result of Grama et al. [Stochastic Process. Appl., 127(4), 1255-1281, 2017]. Moreover, an exponential nonuniform Berry-Esseen bound is also given. At last, some applications of the main results to the confidence interval estimation for the criticality parameter and the population size $Z_n$ are discussed.