Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T06:39:03.619Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Functional Central Limit Theorem for Reversible Markov Chains with Nonlinear Growth of the Variance

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Martial Longla ,
Costel Peligrad  and
Magda Peligrad
Martial Longla*
Affiliation:
University of Cincinnati
Costel Peligrad*
Affiliation:
University of Cincinnati
Magda Peligrad*
Affiliation:
University of Cincinnati
*
∗ Postal address: Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA.
∗ Postal address: Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA.
∗ Postal address: Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

Keywords

Maximal inequalityreversible processmartingale approximationMarkov chaintightnessfunctional central limit theorem

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60G05: Foundations of stochastic processes 60G10: Stationary processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1091 - 1105
Copyright
© Applied Probability Trust 

Footnotes

Supported in part by a Charles Phelps Taft Memorial Fund grant, the NSA grant H98230-11-1-0135, and the NSF grant DMS-1208237.

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bradley, R. C. (2007a). Introduction to Strong Mixing Conditions, Vol. 1. Kendrick Press, Heber City, UT.Google Scholar
Bradley, R. C. (2007b). Introduction to Strong Mixing Conditions, Vol. 2. Kendrick Press, Heber City, UT.Google Scholar
Cuny, C. and Peligrad, M. (2012). Central limit theorem started at a point for stationary processes and additive functionals of reversible Markov chains. J. Theoret. Prob. 25, 171–188.Google Scholar
De La Peña, V. and Giné, E. (1999). Decoupling. From Dependence to Independence. Randomly Stopped Processes. U-Statistics and Processes. Martingales and Beyond. Springer, New York.Google Scholar
Derriennic, Y. and Lin, M. (2001). The central limit thorem for Markov chains with normal transition operators, started at a point. Prob. Theory Relat. Fields 119, 508–528.CrossRefGoogle Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Doukhan, P., Massart, P. and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Prob. Statist. 30, 63–82.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, New York.Google Scholar
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.Google Scholar
Lyons, T. J. and Zheng, W. A. (1988). A crossing estimate for the canonical process on a Dirichlet space and a tightness result. In Colloque Paul Lévy sur les Processes Stochastiques (Palaiseau, 1987; Astérisque 157–158), pp. 249–272.Google Scholar
Merlevède, F. and Peligrad, M. (2012). Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples. To appear in Ann. Prob. Google Scholar
Meyer, P. A. and Zheng, W. A. (1984). Construction du processus de Nelson reversible. In Séminaire de Probabilités XIX (Lecture Notes Math. 1123), Springer, Berlin, pp. 12–26.Google Scholar
Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants (Math. Appl. 31). Springer, Berlin.Google Scholar
Rio, E. (2009). Moment inequalities for sums of dependent random variables under projective conditions. J. Theoret. Prob. 22, 146–163.Google Scholar
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22, 1701–1762.Google Scholar
Wu, L. (1999). Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes. Ann. Inst. H. Poincaré Prob. Statist. 35, 121–141.Google Scholar
Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Prob. 32, 1674–1690.CrossRefGoogle Scholar
Zhao, O. and Woodroofe, M. (2008). Law of the iterated logarithm for stationary processes. Ann. Prob. 36, 127–142.Google Scholar
Zhao, O., Woodroofe, M. and Volný, D. (2010). A central limit theorem for reversible processes with nonlinear growth of variance. J. Appl. Prob. 47, 1195–1202.Google Scholar