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

Part of: Limit theorems Stochastic processes

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.
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Abstract

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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.

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