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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 9 open journal systems 


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.


Full text: PDF

Pages: 85-98

Published on: February 5, 2008
Bibliography
  1. Athreya, K. B.; Ney, P. A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 (1978), 493--501. MR0511425 (80i:60092)
  2. Bradley, Richard C., Jr. Information regularity and the central limit question. Rocky Mountain J. Math. 13 (1983), no. 1, 77--97. MR0692579 (84m:60046)
  3. Breyer, L. A.; Roberts, G. O. Catalytic perfect simulation. Methodol. Comput. Appl. Probab. 3 (2001), no. 2, 161--177. MR1868568 (2002i:60103)
  4. 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. Sttist. and Prob. 2 458--512.
  5. Geyer C. J. (1992). Practical Markov Chain Monte Carlo. Stat. Sci. 7, 473--511.
  6. Häggström, Olle. On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Related Fields 132 (2005), no. 1, 74--82. MR2136867 (2005m:60155)
  7. Hobert, James P.; Robert, Christian P. A mixture representation of $pi$ with applications in Markov chain Monte Carlo and perfect sampling. Ann. Appl. Probab. 14 (2004), no. 3, 1295--1305. MR2071424 (2005d:60106)
  8. Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables.With a supplementary chapter by I. A. Ibragimov and V. V. Petrov.Translation from the Russian edited by J. F. C. Kingman.Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp. MR0322926 (48 #1287)
  9. Jones, Galin L. On the Markov chain central limit theorem. Probab. Surv. 1 (2004), 299--320 (electronic). MR2068475 (2005j:60137)
  10. Jones, Galin L.; Haran, Murali; Caffo, Brian S.; Neath, Ronald. Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101 (2006), no. 476, 1537--1547. MR2279478
  11. Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability.Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6 MR1287609 (95j:60103)
  12. Nummelin, E. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 (1978), no. 4, 309--318. MR0501353 (58 #18732)
  13. Nummelin, Esa. General irreducible Markov chains and nonnegative operators.Cambridge Tracts in Mathematics, 83. Cambridge University Press, Cambridge, 1984. xi+156 pp. ISBN: 0-521-25005-6 MR0776608 (87a:60074)
  14. Nummelin E. (2002). MC's for MCMC'ists. International Statistical Review. 70 215--240.
  15. Propp, James Gary; Wilson, David Bruce. Exact sampling with coupled Markov chains and applications to statistical mechanics.Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995). Random Structures Algorithms 9 (1996), no. 1-2, 223--252. MR1611693 (99k:60176)
  16. Roberts, Gareth O.; Rosenthal, Jeffrey S. Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 (1997), no. 2, 13--25 (electronic). MR1448322 (99b:60122)
  17. Roberts, Gareth O.; Rosenthal, Jeffrey S. General state space Markov chains and MCMC algorithms. Probab. Surv. 1 (2004), 20--71 (electronic). MR2095565 (2005i:60135)













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Electronic Communications in Probability. ISSN: 1083-589X