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

Statistics [?] Probability Letters (Impact Factor: 0.53). 01/2008; 78(12):1649-1655. DOI: 10.1016/j.spl.2008.01.021
Source: RePEc

ABSTRACT Let (Xn) be a Markov chain on measurable space with unique stationary distribution [pi]. Let be a measurable function with finite stationary mean . 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 [pi](h2+[delta])<[infinity], [delta]>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 [pi](h2)<[infinity] 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.

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    ABSTRACT: In the thesis we take the split chain approach to analyzing Markov chains and use it to establish fixed-width results for estimators obtained via Markov chain Monte Carlo procedures (MCMC). Theoretical results include necessary and sufficient conditions in terms of regeneration for central limit theorems for ergodic Markov chains and a regenerative proof of a CLT version for uniformly ergodic Markov chains with $E_{\pi}f^2< \infty.$ To obtain asymptotic confidence intervals for MCMC estimators, strongly consistent estimators of the asymptotic variance are essential. We relax assumptions required to obtain such estimators. Moreover, under a drift condition, nonasymptotic fixed-width results for MCMC estimators for a general state space setting (not necessarily compact) and not necessarily bounded target function $f$ are obtained. The last chapter is devoted to the idea of adaptive Monte Carlo simulation and provides convergence results and law of large numbers for adaptive procedures under path-stability condition for transition kernels. Comment: PhD thesis, University of Warsaw, supervisor - Wojciech Niemiro
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