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On the invariance principle for reversible Markov chains

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  21 June 2016

Magda Peligrad  and
Sergey Utev
Magda Peligrad*
Affiliation:
University of Cincinnati
Sergey Utev*
Affiliation:
University of Leicester
*
* Postal address: Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA. Email address: [email protected]
** Postal address: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK. Email address: [email protected]
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Abstract

In this paper we investigate the functional central limit theorem (CLT) for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional CLT is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.

Keywords

Reversible processMarkov chainfunctional central limit theorem

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60J05: Discrete-time Markov processes on general state spaces
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 593 - 599
Copyright
Copyright © Applied Probability Trust 2016 

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