Electronic Communications in Probability

A Regeneration Proof of the Central Limit Theorem for Uniformly Ergodic Markov Chains

Witold Bednorz, Krzysztof Latuszynski, and Rafal Latala

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Central limit theorems for functionals of general state space Markov chains are of crucial importance in sensible implementation of Markov chain Monte Carlo algorithms as well as of vital theoretical interest. Different approaches to proving this type of results under diverse assumptions led to a large variety of CLT versions. However due to the recent development of the regeneration theory of Markov chains, many classical CLTs can be reproved using this intuitive probabilistic approach, avoiding technicalities of original proofs. In this paper we provide a characterization of CLTs for ergodic Markov chains via regeneration and then use the result to solve the open problem posed in [Roberts & Rosenthal 2005]. We then discuss the difference between one-step and multiple-step small set condition.

Article information

Electron. Commun. Probab. Volume 13 (2008), paper no. 9, 85-98.

Accepted: 24 January 2008
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60F05: Central limit and other weak theorems

Markov chains central limit theorems regeneration ergodicity uniform ergodicity Harris recurrence

This work is licensed under a Creative Commons Attribution 3.0 License.


Bednorz, Witold; Latuszynski, Krzysztof; Latala, Rafal. A Regeneration Proof of the Central Limit Theorem for Uniformly Ergodic Markov Chains. Electron. Commun. Probab. 13 (2008), paper no. 9, 85--98. doi:10.1214/ECP.v13-1354. https://projecteuclid.org/euclid.ecp/1465233436

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