A regeneration proof of the central limit theorem for uniformly ergodic Markov chains

Let (Xn) be a Markov chain on measurable space (E, E) with unique stationary distribution π. Let h : E → R be a measurable function with finite stationary mean π (h) {colon equals} ∫E h (x) π (d x). Ibragimov and Linnik [Ibragimov, I.A., Linnik, Y.V., 1971. Independent and Stationary Sequences of Random Variables. Wolter-Noordhoff, Groiningen] proved that if (Xn) is geometrically ergodic, then a central limit theorem (CLT) holds for h whenever π (| h |2 + δ) < ∞, δ > 0. Cogburn [Cogburn, R., 1972. The central limit theorem for Markov processes. In: Le Cam, L.E., Neyman, J., Scott, E.L. (Eds.), Proc. Sixth Ann. Berkley Symp. Math. Statist. and Prob., 2. pp. 485-512] proved that if a Markov chain is uniformly ergodic, with π (h2) < ∞ then a CLT holds for h. The first result was re-proved in Roberts and Rosenthal [Roberts, G.O., Rosenthal, J.S., 2004. General state space Markov chains and MCMC algorithms. Prob. Surveys 1, 20-71] using a regeneration approach; thus removing many of the technicalities of the original proof. This raised an open problem: to provide a proof of the second result using a regeneration approach. In this paper we provide a solution to this problem. © 2008 Elsevier B.V. All rights reserved.