A Regeneration Proof of the Central Limit Theorem for Uniformly
Ergodic Markov Chains
Witold Bednorz, Warsaw University
Krzysztof Latuszynski, Warsaw School of Economics
Rafal Latala, Warsaw University
Abstract
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.
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