A Regeneration Proof of the Central Limit Theorem for Uniformly Ergodic Markov Chains

Witold Bednorz,Krzysztof Latuszynski,Rafal Latala

doi:10.1214/ecp.v13-1354

Abstract

Central limit theorems for functionals of general state space Markov chains are of crucial importance in sensible implementation of Markov chain Monte Carlo algorithms as well as of vital theoretical interest. Different approaches to proving this type of results under diverse assumptions led to a large variety of CLT versions. However due to the recent development of the regeneration theory of Markov chains, many classical CLTs can be reproved using this intuitive probabilistic approach, avoiding technicalities of original proofs. In this paper we provide a characterization of CLTs for ergodic Markov chains via regeneration and then use the result to solve the open problem posed in [Roberts & Rosenthal 2005]. We then discuss the difference between one-step and multiple-step small set condition.

Highlights

IntroductionLet (Xn)n≥0 be a time homogeneous Markov chain on a measurable space (X, B(X)) with initial distribution π0, transition kernel P and a unique stationary gn
Let (Xn)n≥0 be a time homogeneous Markov chain on a measurable space (X, B(X)) with initial distribution π0, transition kernel P and a unique stationary gn =1 n distribution π.n−1 i=0 g(Xi) and Let Eπ g g be =a real valued Borel function X g(x)π(dx)
We say that a Markov chain (Xn)n≥0 with transition kernel P and stationary distribution π is geometrically ergodic, if P n(x, ·) − π(·) tv ≤ M (x)ρn, for some ρ < 1

Summary

Introduction

Let (Xn)n≥0 be a time homogeneous Markov chain on a measurable space (X, B(X)) with initial distribution π0, transition kernel P and a unique stationary gn. We state two classical CLT versions for geometrically ergodic and uniformly ergodic Markov chains. Let μ1(·) − μ2(·) tv := 2 supA∈B |μ1(A) − μ2(A)| be the well known total variation distance between probability measures μ1 and μ2. We say that a Markov chain (Xn)n≥0 with transition kernel P and stationary distribution π is geometrically ergodic, if P n(x, ·) − π(·) tv ≤ M (x)ρn, for some ρ < 1. 15thEYSM Castro Urdiales (Spain) and M (x) < ∞ π−almost everywhere. We say it is uniformly ergodic, if P n(x, ·) − π(·) tv ≤ M ρn, for some ρ < 1 and M < ∞

If a ly ergodic

Small Sets and the Split Chain

A Proof