Exponential approximation for the nearly critical Galton-Watson process and occupation times of Markov chains

TL;DR

This paper applies Stein's method to provide exponential approximation error bounds for nearly critical Galton-Watson processes conditioned on non-extinction and for occupation times of Markov chains, including a new rate for revisits in 2D random walks.

Contribution

It introduces new exponential approximation bounds for these stochastic processes, extending previous work with novel applications and improved rates, especially for 2D random walk revisits.

Findings

Error bounds for nearly critical Galton-Watson process conditioned on non-extinction.

New exponential approximation rate for occupation times in Markov chains.

Improved rate for revisits to the origin in 2D random walks (Erdős-Taylor theorem).

Abstract

In this article we provide new applications for exponential approximation using the framework of Pek\"oz and R\"ollin (in press), which is based on Stein's method. We give error bounds for the nearly critical Galton-Watson process conditioned on non-extinction, and for the occupation times of Markov chains; for the latter, in particular, we give a new exponential approximation rate for the number of revisits to the origin for general two dimensional random walk, also known as the Erd\H{o}s-Taylor theorem.