Exponential Approximation for the Nearly Critical Galton-Watson Process and Occupation Times of Markov Chains

Erol Peköz,Adrian Röllin

doi:10.1214/ejp.v16-914

Abstract

In this article we provide new applications for exponential approximation using the framework of Pekoz and Rollin (2011), 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 Erdos-Taylor theorem.

Highlights

A new framework for estimating the error of the exponential approximation was recently developed in Peköz and Röllin (2011), where it was applied to geometric sums, Markov chain hitting times, and the critical Galton-Watson conditioned on non-extinction
In this article we provide some generalizations to the approach of Peköz and Röllin (2011) and apply them to study Markov chain occupation times and a result of Erdos and Taylor (1960) for the number of visits to the origin by the two dimensional random walk, as well as to get a rate for the result of Fahady, Quine, and Vere-Jones (1971) for the nearly critical Galton-Watson branching process conditioned on non-extinction
The main result in Peköz and Röllin (2011) that we use is based on Stein’s method (see e.g. Ross and Peköz (2007) for an introduction) and can be thought of as formalizing the intuitive notion that a random variable X has approximately an exponential distribution if X and X e are close in distribution, where X e has the equilibrium distribution with respect to X characterized by

Summary

INTRODUCTION

A new framework for estimating the error of the exponential approximation was recently developed in Peköz and Röllin (2011), where it was applied to geometric sums, Markov chain hitting times, and the critical Galton-Watson conditioned on non-extinction. In this article we provide some generalizations to the approach of Peköz and Röllin (2011) and apply them to study Markov chain occupation times and a result of Erdos and Taylor (1960) for the number of visits to the origin by the two dimensional random walk, as well as to get a rate for the result of Fahady, Quine, and Vere-Jones (1971) for the nearly critical Galton-Watson branching process conditioned on non-extinction. We can obtain a bound on the error in terms of how closely X and U X s can be coupled, where U is an independent uniform (0,1) random variable independent of all else; X e has the same distribution as U X s This last approach is the one we use below for the nearly critical Galton Watson process conditioned on non-extinction.

PRELIMINARIES

THE NEARLY CRITICAL GALTON-WATSON BRANCHING PROCESS

VISITS TO THE ORIGIN FOR A TWO DIMENSIONAL SIMPLE RANDOM WALK