Electronic Communications in Probability

Asymptotic variance of functionals of discrete-time Markov chains via the Drazin inverse.

Dan Spitzner and Thomas Boucher

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We consider a $\psi$-irreducible, discrete-time Markov chain on a general state space with transition kernel $P$. Under suitable conditions on the chain, kernels can be treated as bounded linear operators between spaces of functions or measures and the Drazin inverse of the kernel operator $I - P$ exists. The Drazin inverse provides a unifying framework for objects governing the chain. This framework is applied to derive a computational technique for the asymptotic variance in the central limit theorems of univariate and higher-order partial sums. Higher-order partial sums are treated as univariate sums on a `sliding-window' chain. Our results are demonstrated on a simple AR(1) model and suggest a potential for computational simplification.

Article information

Electron. Commun. Probab., Volume 12 (2007), paper no. 13, 120-133.

Accepted: 24 April 2007
First available in Project Euclid: 6 June 2016

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General state space Markov chains $f$-regularity Markov chain central limit theorem Drazin inverse fundamental matrix asymptotic variance

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Spitzner, Dan; Boucher, Thomas. Asymptotic variance of functionals of discrete-time Markov chains via the Drazin inverse. Electron. Commun. Probab. 12 (2007), paper no. 13, 120--133. doi:10.1214/ECP.v12-1262. https://projecteuclid.org/euclid.ecp/1465224956

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  • Brockwell, Peter J.; Davis, Richard A. Time series: theory and methods. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1991. xvi+577 pp. ISBN: 0-387-97429-6
  • Campbell, Stephen L. Continuity of the Drazin inverse. Linear and Multilinear Algebra 8 (1979/80), no. 3, 265–268.
  • Campbell, Stephen L.; Meyer, C. D., Jr. Generalized inverses of linear transformations. Surveys and Reference Works in Mathematics, 4. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. xi+272 pp. ISBN: 0-273-08422-4
  • Chauveau, Didier; Diebolt, Jean. Estimation of the asymptotic variance in the CLT for Markov chains. Stoch. Models 19 (2003), no. 4, 449–465.
  • Tierney, Luke. Markov chains for exploring posterior distributions. With discussion and a rejoinder by the author. Ann. Statist. 22 (1994), no. 4, 1701–1762.
  • R. Cogburn. The central limit theorem for Markov processes. Proceedings of the 6th Berkeley symposium on mathematical statistics and probability, (1972), 485-512.
  • Djordjević, D. S.; Stanimirović, P. S.; Wei, Y. The representation and approximations of outer generalized inverses. Acta Math. Hungar. 104 (2004), no. 1-2, 1–26.
  • François, Olivier. On the spectral gap of a time reversible Markov chain. Probab. Engrg. Inform. Sci. 13 (1999), no. 1, 95–101.
  • C. J. Geyer. Practical Markov chain Monte Carlo (with discussion). Statistical Science, (1992), 7:473-511.
  • Castro González, N.; Koliha, J. J. Semi-iterative methods for the Drazin inverse solution of linear equations in Banach spaces. Numer. Funct. Anal. Optim. 20 (1999), no. 5-6, 405–418.
  • Glynn, Peter W.; Meyn, Sean P. A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 24 (1996), no. 2, 916–931.
  • Hobert, James P.; Jones, Galin L.; Presnell, Brett; Rosenthal, Jeffrey S. On the applicability of regenerative simulation in Markov chain Monte Carlo. Biometrika 89 (2002), no. 4, 731–743.
  • Jones, Galin L. On the Markov chain central limit theorem. Probab. Surv. 1 (2004), 299–320 (electronic).
  • Kemeny, John G.; Snell, J. Laurie. Finite Markov chains. The University Series in Undergraduate Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York 1960 viii+210 pp.
  • Kipnis, C.; Varadhan, S. R. S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986), no. 1, 1–19.
  • Koliha, J. J. A generalized Drazin inverse. Glasgow Math. J. 38 (1996), no. 3, 367–381.
  • Koliha, J. J. Isolated spectral points. Proc. Amer. Math. Soc. 124 (1996), no. 11, 3417–3424.
  • Koliha, J. J.; Rakočević, V. Continuity of the Drazin inverse. II. Studia Math. 131 (1998), no. 2, 167–177.
  • Lawler, Gregory F.; Sokal, Alan D. Bounds on the $L^2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309 (1988), no. 2, 557–580.
  • León, Carlos A. Maximum asymptotic variance of sums of finite Markov chains. Statist. Probab. Lett. 54 (2001), no. 4, 413–415.
  • Meyer, Carl D., Jr. The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17 (1975), 443–464.
  • 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
  • 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
  • Nummelin, E. (2002), MC's for MCMC'ists, International Statistical Review, 70:215-240.
  • Peskun, P. H. Optimum Monte-Carlo sampling using Markov chains. Biometrika 60 (1973), 607–612.
  • Robert, Christian P. Convergence control methods for Markov chain Monte Carlo algorithms. Statist. Sci. 10 (1995), no. 3, 231–253.
  • Robert, Christian P.; Casella, George. Monte Carlo statistical methods. Springer Texts in Statistics. Springer-Verlag, New York, 1999. xxii+507 pp. ISBN: 0-387-98707-X
  • Roberts, G. O.; Tweedie, R. L. Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 (1996), no. 1, 95–110.
  • Roberts, Gareth O.; Rosenthal, Jeffrey S. General state space Markov chains and MCMC algorithms. Probab. Surv. 1 (2004), 20–71 (electronic).
  • Tierney, Luke. Markov chains for exploring posterior distributions. With discussion and a rejoinder by the author. Ann. Statist. 22 (1994), no. 4, 1701–1762.
  • Tóth, Bálint. Persistent random walks in random environment. Probab. Theory Relat. Fields 71 (1986), no. 4, 615–625.