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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, Sn say, of a reversible Markov chain, the condition E[Sn2] / n → κ ∈ (0, ∞) implies the convergence of the conditional distribution of Sn / √E[Sn2], given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, E[Sn2] = nl(n), where l is a slowly varying function. It is shown by example that the conditional distributions of Sn / √E[Sn2] 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
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1195 - 1202
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
[2] Bingham, N. H., Goldie, C. M. and Tuegels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
[3] Cuny, C. and Peligrad, M. (2010). Central limit theorem started at a point for stationary processes and additive functionals of reversible Markov chains. To appear in J. Theoret. Prob. Available at http://www.springerlink.com/content/f888474451015636/fulltext.pdf.Google Scholar
[4] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[5] Halmos, P. (1957). Introduction to Hilbert Spaces and the Theory of Spectral Multiplicity, 2nd edn. Chelsea, New York.Google Scholar
[6] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin.Google Scholar
[7] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104, 1–19.CrossRefGoogle Scholar
[8] Koul, H. L. and Surgailis, D. (2001). Asymptotics of empirical processes of long memory moving averages with infinite variance. Stoch. Process. Appl. 91, 309–336.Google Scholar
[9] Loève, M. (1977). Probability Theory. I, 4th edition, Springer, New York.Google Scholar
[10] Merlevède, F. and Peligrad, M. (2006). On the weak invariance principle for stationary sequences under projective criteria. J. Theoret. Prob. 19, 647–689.Google Scholar
[11] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants (Math. Appl. 31). Springer, Berlin.Google Scholar
[12] Tierney, L. (1994). Markov chains for exploring posterior distribution. Ann. Statist. 22, 1701–1762.Google Scholar
[13] Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Prob. 32, 1674–1690.Google Scholar
[14] Zhao, O. and Woodroofe, M. (2008). On martingale approximations. Ann. Appl. Prob. 18, 1831–1847.Google Scholar