Probability Surveys

On the Markov chain central limit theorem

Galin L. Jones

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Abstract

The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.

Article information

Source
Probab. Surveys Volume 1 (2004), 299-320.

Dates
First available in Project Euclid: 29 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.ps/1104335301

Digital Object Identifier
doi:10.1214/154957804100000051

Mathematical Reviews number (MathSciNet)
MR2068475

Zentralblatt MATH identifier
1189.60129

Keywords
Central Limit Theorem Markov Chain Monte Carlo Mixing Condition Drift Condition

Citation

Jones, Galin L. On the Markov chain central limit theorem. Probab. Surveys 1 (2004), 299--320. doi:10.1214/154957804100000051. https://projecteuclid.org/euclid.ps/1104335301.


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References

  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bradley, R. C. (1983). Information inequality and the central limit question. Rocky Mountain Journal of Mathematics 13: 77-97.
  • Bradley, R. C. (1985). On the central limit question under absolute regularity. The Annals of Probability, 13:1314--1325.
  • Bradley, R. C. (1986). Basic properties of strong mixing conditions. In Eberlein, E. and Taqqu, M. S., editors, Dependence in Probability and Statistics: A Survey of Recent Results, pages 165--192. Birkhauser, Cambridge, MA.
  • Bradley, R. C. (1999). Can a theorem of Csáki and Fischer provide a key to Ibragimov's conjecture? In Bolyai Society Mathematical Studies 9, Random Walks, Budapest (Hungary) 1999, pages 11--42. North-Holland, Amsterdam.
  • Chan, K. S. and Geyer, C. J. (1994). Comment on ``Markov chains for exploring posterior distributions''. The Annals of Statistics, 22:1747--1758.
  • Chen, X. (1999). Limit theorems for functionals of ergodic Markov chains with general state space. Memoirs of the American Mathematical Society, 139.
  • Christensen, O. F., Moller, J., and Waagepetersen, R. P. (2001). Geometric ergodicity of Metropolis-Hastings algorithms for conditional simulation in generalized linear mixed models. Methodology and Computing in Applied Probability, 3:309--327.
  • Cogburn, R. (1960). Asymptotic properties of stationary sequences. University of California Publications in Statistics, 30, 63--82.
  • Cogburn, R. (1972). The central limit theorem for Markov processes. In Le Cam, L. E., Neyman, J., and Scott, E. L., editors, Proceedings of the Sixth Annual Berkeley Symposium on Mathematical Statistics and Probability, volume 2, pages 485--512. University of California Press, Berkeley.
  • Damerdji, H. (1994). Strong consistency of the variance estimator in steady-state simulation output analysis. Mathematics of Operations Research, 19:494--512.
  • Davydov, Y. A. (1973). Mixing conditions for Markov chains. Theory of Probability and Its Applications, 27:312--328.
  • Dehling, H., Denker, M., and Philipp, W. (1986). Central limit theorems for mixing sequences of random variables under minimal conditions. The Annals of Probability, 14:1359--1370.
  • Denker, M. (1986). Uniform integrability and the central limit theorem. In Eberlein, E. and Taqqu, M. S., editors, Dependence in Probability and Statistics: A Survey of Recent Results, pages 269--274. Birkhauser, Cambridge, MA.
  • Doob, J. L. (1953). Stochastic Processes. John Wiley & Sons, New York.
  • Douc, R., Fort, G., Moulines, E., and Soulier, P. (2204). Practical drift conditions for subgeometric rates of convergence. The Annals of Applied Probability, 14:1353--1377.
  • Doukhan, P., Massart, P., and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Annals of the Institute Henri Poincaré, Probability and Statistics, 30: 63-82.
  • Fort, G. and Moulines, E. (2000). V-subgeometric ergodicity for a Hastings-Metropolis algorithm. Statistics and Probability Letters, 49:401--410.
  • Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Processes and their Applications, 103:57--99.
  • Gaver, D. P. and O'Muircheartaigh, I. G. (1987). Robust empirical Bayes analyses of event rates. Technometrics, 29:1--15.
  • Geyer, C. J. (1992). Practical Markov chain Monte Carlo (with discussion). Statistical Science, 7:473--511.
  • Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Barndorff-Nielsen, O. E., Kendall, W. S., and van Lieshout, M. N. M., editors, Stochastic Geometry: Likelihood and Computation, pages 79--140. Chapman & Hall/CRC, Boca Raton.
  • Glynn, P. W. and Whitt, W. (1992). The asymptotic validity of sequential stopping rules for stochastic simulations. The Annals of Applied Probability, 2:180--198.
  • Häggström, O. (2004). On the central limit theorem for geometrically ergodic Markov chains. Probability Theory and Related Fields (to appear).
  • Herndorf, N. (1983). Stationary strongly mixing sequences not satisfying the central limit theorem. The Annals of Probability, 11:809--813.
  • Hobert, J. P. and Geyer, C. J. (1998). Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model. Journal of Multivariate Analysis, 67:414--430.
  • Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Theory of Probability and Its Applications, 7:349--382.
  • Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Walters-Noordhoff, The Netherlands.
  • Ibragimov, I. A., (1975) A note on the central limit theorem for dependent random variables. Theory of Probability and its Applications 20: 135-141.
  • Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Processes and Their Applications, 85:341--361.
  • Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. The Annals of Applied Probability, 12:224--247.
  • Jones, G. L., Haran, M., and Caffo, B. S. (2004). Output analysis for Markov chain Monte Carlo simulations. Technical report, University of Minnesota, School of Statistics.
  • Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statistical Science, 16:312--334.
  • Jones, G. L. and Hobert, J. P. (2004). Sufficient burn-in for Gibbs samplers for a hierarchical random effects model. The Annals of Statistics, 32:784--817.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Communications in Mathematical Physics, 104:1--19.
  • Kontoyiannis, I. and Meyn, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. The Annals of Applied Probability, 13:304--362.
  • Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
  • Liu, J. S., Wong, W. H., and Kong, A. (1995). Covariance structure and convergence rate of the Gibbs sampler with various scans. Journal of the Royal Statistical Society, Series B, 57:157--169.
  • Marchev, D. and Hobert, J. P. (2004). Geometric ergodicity of van Dyk and Meng's algorithm for the multivariate Student's $t$ model. Journal of the American Statistical Association, 99:228--238.
  • Mengersen, K. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. The Annals of Statistics, 24:101--121.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer-Verlag, London.
  • Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. The Annals of Applied Probability, 4:981--1011.
  • Mira, A. and Tierney, L. (2002). Efficiency and convergence properties of slice samplers. Scandinavian Journal of Statistics, 29:1--12.
  • Møller, J. (1999). Markov chain Monte Carlo and spatial point processes. In Barndorff-Nielsen, O. E., Kendall, W. S., and van Lieshout, M. N. M., editors, Stochastic Geometry: Likelihood and Computation, pages 141--172. Chapman & Hall/CRC, Boca Raton.
  • Mori, T. and Yoshihara, K. (1986). A note on the central limit theorem for stationary strong mixing sequences. Yokohama Mathematical Journal, 34: 143--146.
  • Nummelin, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press, London.
  • Nummelin, E. (2002). MC's for MCMC'ists. International Statistical Review, 70:215--240.
  • Peligrad, M. (1982). Invariance principles for mixing sequences of random variables. The Annals of Probability, 10:968--981.
  • Philipp, W. and Stout, W. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs of the American Mathematical Society, 2:1--140.
  • Robert, C. P. (1995). Convergence control methods for Markov chain Monte Carlo algorithms. Statistical Science, 10:231--253.
  • Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.
  • Roberts, G. O. (1999). A note on acceptance rate criteria for CLTs for Metropolis-Hastings algorithms. Journal of Applied Probability, 36:1210--1217.
  • Roberts, G. O. and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. Journal of the Royal Statistical Society, Series B, 56:377--384.
  • Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability, 2:13--25.
  • Roberts, G. O. and Rosenthal, J. S. (1999). Convergence of slice sampler Markov chains. Journal of the Royal Statistical Society, Series B, 61:643--660.
  • Roberts, G. O. and Rosenthal, J. S. (2001). Markov chains and de-initializing processes. Scandinavian Journal of Statistics, 28:489--504.
  • Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probability Surveys, 1:20--71.
  • Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika, 83:95--110.
  • Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Springer-Verlag, New York.
  • Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association, 90:558--566.
  • Rosenthal, J. S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimators. Statistics and Computing, 6:269--275.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). The Annals of Statistics, 22:1701--1762.
  • Tuomimem, P. and Tweedie, R. L. (1994). Subgeometric rates of convergence of $f$-ergodic Markov chains. Advances in Applied Probability, 26:775--798.