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

Let (Xn)(Xn) be a Markov chain on measurable space (E,E)(E,E) with unique stationary distribution ππ. Let h:E→Rh:E→R be a measurable function with finite stationary mean π(h)≔∫Eh(x)π(dx). Ibragimov and Linnik [Ibragimov, I.A., Linnik, Y.V., 1971. Independent and Stationary Sequences of Random Variables. Wolter-Noordhoff, Groiningen] proved that if (Xn)(Xn) is geometrically ergodic, then a central limit theorem (CLT) holds for hh whenever π(|h|2+δ)<∞π(|h|2+δ)<∞, δ>0δ>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)<∞π(h2)<∞ then a CLT holds for hh. 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.