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A regeneration proof of the central limit theorem for uniformly ergodic Markov chains. (English) Zbl 1194.60046

Summary: 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 by G. O. Roberts and J. S. Rosenthal [Probab. Surv. 1, 20–71, electronic only (2004; Zbl 1189.60131)]. We then discuss the difference between one-step and multiple-step small set condition.

MSC:

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

Citations:

Zbl 1189.60131
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