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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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References

Billingsley, P. (1995).Probability and Measure, 3rd edn.John Wiley, New York.Google Scholar
Deligiannidis, G., Peligrad, M. and Utev, S. (2015).Asymptotic variance of stationary reversible and normal Markov processes.Electron. J. Prob. 20, 26pp.Google Scholar
Giraudo, D. and Volný, D. (2014).A strictly stationary β-mixing process satisfying the central limit theorem but not the weak invariance principle.Stoch. Process. Appl. 124, 3769–3781.CrossRefGoogle Scholar
Gordin, M. I. and Lifšic, B. (1981).A remark about a Markov process with normal transition operator. In Proc. Third Vilnius Conference of Probability and Statistics, Vol. 1, Akad. Nauk Litovsk, Vilnius, pp.147–148 (in Russian).Google Scholar
Kipnis, C. and Landim, C. (1999).Scaling Limits of Interacting Particle Systems.Springer, Berlin.CrossRefGoogle 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.CrossRefGoogle Scholar
Longla, M., Peligrad, C. and Peligrad, M. (2012).On the functional central limit theorem for reversible Markov chains with nonlinear growth of the variance.J. Appl. Prob. 49, 1091–1105.Google Scholar
Skorokhod, A. V. (1956).Limit theorems for stochastic processes.Theory Prob. Appl. 1, 261–290.CrossRefGoogle Scholar
Tierney, L. (1994).Markov chains for exploring posterior distribution.Ann. Statist. 22, 1701–1762.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