Chernoff-type bound for finite Markov chains
Pascal Lezaud
Source: Ann. Appl. Probab. Volume 8, Number 3
(1998), 849-867.
Abstract
This paper develops bounds on the distribution function of the empirical mean for irreducible finite-state Markov chains. One approach, explored by Gillman, reduces this problem to bounding the largest eigenvalue of a perturbation of the transition matrix for the Markov chain. By using estimates on eigenvalues given in Kato's book Perturbation Theory for Linear Operators, we simplify the proof of Gillman and extend it to nonreversible finite-state Markov chains and continuous time. We also set out another method, directly applicable to some general ergodic Markov kernels having a spectral gap.
First Page:
Show
Hide
Primary Subjects:
60F10
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1028903453
Mathematical Reviews number (MathSciNet): MR1627795
Digital Object Identifier: doi:10.1214/aoap/1028903453
Zentralblatt MATH identifier: 0938.60027
References
Aldous, D. (1987). On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Prob. Engrg. Inform. Sci. 1 33-46. Aldous, D. and Fill, J. Reversible Markov chains and random walks on graphs. Unpublished manuscript.
Bahadur, R. R. and Ranga Rao, R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015-1033.
Mathematical Reviews (MathSciNet): MR22:8549
Zentralblatt MATH: 0101.12603
Digital Object Identifier: doi:10.1214/aoms/1177705674
Project Euclid: euclid.aoms/1177705674
Bennett, G. (1962). Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc. 57 33-45.
Chernoff, H. (1952). A measure of asy mptotic efficiency for tests of a hy pothesis based on the sum of observations. Ann. Math. Statist. 23 493-507.
Mathematical Reviews (MathSciNet): MR15,241c
Zentralblatt MATH: 0048.11804
Digital Object Identifier: doi:10.1214/aoms/1177729330
Project Euclid: euclid.aoms/1177729330
Cram´er, H. (1938). Sur un nouveau th´eor eme de la th´eorie des probabilit´es. Actualit´es Sci. Indust. 736.
Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston.
Mathematical Reviews (MathSciNet): MR95a:60034
Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696-730.
Mathematical Reviews (MathSciNet): MR94i:60074
Zentralblatt MATH: 0799.60058
Digital Object Identifier: doi:10.1214/aoap/1177005359
Project Euclid: euclid.aoap/1177005359
Diaconis, P. and Saloff-Coste, L. (1995). What do we know about the Metropolis algorithm? Unpublished manuscript. Diaconis, P. and Saloff-Coste, L. (1996a). Logarithmic Sobolev inequalities and finite Markov chains. Ann. Appl. Probab. 6 695-750. Diaconis, P. and Saloff-Coste, L. (1996b). Nash inequalities for finite Markov chains. J. Appl. Probab. 9 459-510.
Dinwoodie, I. (1995). A probability inequality for the occupation measure of a reversible Markov chain. Ann. Appl. Probab. 5 37-43.
Mathematical Reviews (MathSciNet): MR96d:60099
Zentralblatt MATH: 0829.60022
Digital Object Identifier: doi:10.1214/aoap/1177004826
Project Euclid: euclid.aoap/1177004826
Fill, J. (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1 62-87.
Mathematical Reviews (MathSciNet): MR92h:60104
Zentralblatt MATH: 0726.60069
Digital Object Identifier: doi:10.1214/aoap/1177005981
Project Euclid: euclid.aoap/1177005981
Gillman, D. (1993). Hidden Markov chains: rates of convergence and the complexity of inference. Ph.D. dissertation, MIT.
Kato, T. (1966). Perturbation Theory for Linear Operators. Springer, New York.
Mathematical Reviews (MathSciNet): MR203473
Kolmogorov, A. (1929). ¨Uber das Gesetz des iterieten Logaritmus. Math. Ann. 99 309-319.
Mann, B. (1996). Berry-Esseen central limit theorem for Markov chains. Ph.D. dissertation, Harvard Univ.
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR81b:00002
Nagaev, S. V. (1957). Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 378-406.
Mathematical Reviews (MathSciNet): MR20:1355
Nagaev, S. V. (1961). More exact statements of limits theorems for homogeneous Markov chains. Theory Probab. Appl. 6 62-81.
Mathematical Reviews (MathSciNet): MR24:A1143
Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR56:13334
Trotter, H. F. (1959). On the product of semi-groups of operators. Proc. Amer. Math. Soc. 10 545-551.
Mathematical Reviews (MathSciNet): MR21:7446
Zentralblatt MATH: 0099.10401
Digital Object Identifier: doi:10.2307/2033649
JSTOR: links.jstor.org
The Annals of Applied Probability
