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A Central Limit Theorem for Reversible Processes with Nonlinear Growth of Variance

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ou Zhao ,
Michael Woodroofe  and
Dalibor Volný
Ou Zhao*
Affiliation:
University of South Carolina
Michael Woodroofe*
Affiliation:
University of Michigan
Dalibor Volný*
Affiliation:
Université de Rouen
*
∗ Postal address: Department of Statistics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, USA. Email address: ouzhao@stat.sc.edu
∗∗ Postal address: Department of Statistics and Mathematics, University of Michigan, 275 West Hall, 1085 South University, Ann Arbor, MI 48109, USA. Email address: michaelw@umich.edu
∗∗∗ Postal address: Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen, France. Email address: dalibor.volny@univ-rouen.fr
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Abstract

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Kipnis and Varadhan (1986) showed that, for an additive functional, S n say, of a reversible Markov chain, the condition E[S n 2] / n → κ ∈ (0, ∞) implies the convergence of the conditional distribution of S n / √E[S n 2], given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, E[S n 2] = n l(n), where l is a slowly varying function. It is shown by example that the conditional distributions of S n / √E[S n 2] need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly nonstandard) normal distribution are developed.

Keywords

Conditional distributionMarkov chainself-adjoint operatorslowly varying function

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60J05: Discrete-time Markov processes on general state spaces

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1195 - 1202
Copyright
Copyright © Applied Probability Trust 2010 

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