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A regeneration proof of the central limit theorem for uniformly

ergodic Markov chains

By

AJAY JASRA

Department of Mathematics, Imperial College London, SW7 2AZ, London, UK

and

CHAO YANG1

Department of Mathematics, University of Toronto, M5S 2E4, Toronto ON, Canada

November 10, 2005

Abstract

Let (Xn) be a Markov chain on measurable space (E,E) with unique stationary distribution

π. Let h : E → R be a measurable function with finite stationary mean π(h) :=?

theorem (CLT) holds for h whenever π(|h|2+δ) < ∞, δ > 0. Cogburn (1972) proved that if a

Markov chain is uniformly ergodic, with π(h2) < ∞ then a CLT holds for h. The first result

was re-proved in Roberts and Rosenthal (2004) using a regeneration approach; thus removing

many of the technicalities of the original proof. This raised an open problem: to provide a proof

of the second result using a regeneration approach. In this paper we provide a solution to this

problem.

Keywords: Markov chains; Central limit theorems

Eh(x)π(dx).

Ibragimov and Linnik (1971) proved that if (Xn) is geometrically ergodic, then a central limit

1 Introduction

Let (Xn) be a Markov chain with transition kernel P : E × E → [0,1] and a unique stationary

distribution π. Let h : E → R be a real-valued measurable function. We say that h satisfies

a Central Limit Theorem (or√n−CLT) if there is some σ2< ∞ such that the normalized sum

n−1

distribution with zero mean and variance σ2(we allow that σ2= 0), and (e.g. Chan and Geyer

(1994), see also Bradley (1985) and Chen (1999))

2?n

i=1[h(Xi)−π(h)] converges weakly to a N(0,σ2) distribution, where N(0,σ2) is a Gaussian

σ2

=π(h2) + 2

?

E

∞

?

n=1

h(x)Pn(h)(x)π(dx)

with Pn(h)(x) =?

measures on (E,E) as P(E). The total variation distance between μ,ν ∈ P(E) is:

Eh(y)Pn(x,dy) and Pn(x,dy) the n−step transition law for the Markov chain.

To further our discussion we provide the following definitions. Denote the class of probability

?μ − ν?

:=sup

A∈E|μ(A) − ν(A)|.

We will be concerned with geometrically and uniformly ergodic Markov chains:

1Corresponding author, e-mail: chaoyang@math.toronto.edu

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Definition 1.1. A Markov chain with stationary distribution π ∈ P(E) is geometrically ergodic if

∀n ∈ N:

?Pn(x,∙) − π(∙)?

?

M(x)ρn

where ρ < 1 and M(x) < ∞ π−almost everywhere. If M = supx∈E|M(x)| is finite then the chain

is uniformly ergodic.

Theorem 1.2 (Cogburn, 1972). If a Markov chain with stationary distribution π ∈ P(E) is

uniformly ergodic, then a√n−CLT holds for h whenever π(h2) < ∞.

Ibragimov and Linnik (1971) proved a CLT for h when the chain is geometrically ergodic and,

for some δ > 0, π(|h|2+δ) < ∞. Roberts and Rosenthal (2004) provided a simpler proof using

regeneration arguments. In addition, Roberts and Rosenthal (2004) left an open problem: To

provide a proof of Theorem 1.2 (originally proved by Cogburn (1972)) using regeneration.

Many of the recent developments of CLTs for Markov chains are related to the evolution of

stochastic simulation algorithms such as Markov chain Monte Carlo (MCMC) (e.g. Robert and

Rosenthal (2004)). For example, Roberts and Rosenthal (2004) posed many open problems, includ-

ing that considered here, for CLTs; see H¨ aggstr¨ om (2005) for a solution to another open problem.

Additionally, Jones (2004) discusses the link between mixing processes and CLTs, with MCMC

algorithms a particular consideration. For an up-to-date review of CLTs for Markov chains see:

Bradley (1985), Chen (1999) and Jones (2004).

The proof of Theorem 1.2, using regeneration theory, provides an elegant framework for the

proof of CLTs for Markov chains. The approach may also be useful for alternative proofs of CLTs

for chains with different ergodicity properties; e.g. polynomial ergodicity (see Jarner and Roberts

(2002)).

The structure of this paper is as follows. In Section 2 we provide some background knowledge

about the small sets and the regeneration construction, we also detail some technical results. In

Section 3 we use the results of the previous Section to provide a proof of Theorem 1.2 using

regenerations.

2Small Sets and Regeneration Construction

2.1 Small Sets

We recall the notion of a small set:

Definition 2.1. A set C ∈ E is small (or (n0,?,ν)-small) if there exists an n0∈ N, ? > 0 and a

non-trivial ν ∈ P(E) such that the following minorization condition holds ∀x ∈ C:

Pn0(x,∙)

?

?ν(∙). (1)

It is known (e.g. Meyn and Tweedie (1993)) that if P is uniformly ergodic, the whole state

space E is small. That is we have the following lemma:

Lemma 2.1. If (Xn) on (E,E) with stationary distribution π ∈ P(E) is uniformly ergodic, then

E is small.

2

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