Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T16:54:16.433Z Has data issue: false hasContentIssue false
  • English
  • Français

Chernoff and Berry–Esséen inequalitiesfor Markov processes

Published online by Cambridge University Press:  15 August 2002

Pascal Lezaud
Pascal Lezaud*
Affiliation:
Centre d'Études de la Navigation Aérienne, 31055 Toulouse Cedex, France; pascal.lezaud@cena.fr. Université Paul Sabatier, 31055 Toulouse Cedex, France.
Get access

Abstract

In this paper, we develop bounds on the distribution function of the empiricalmean for general ergodic Markov processes having a spectral gap. Our approach isbased on the perturbation theory for linear operators, following the techniqueintroduced by Gillman.

Keywords

Markov processChernoff boundBerry–Esséeneigenvaluesperturbation theory.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 183 - 201
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs. Monograph in preparation. Available from the Aldous's home page at http://www.stat.berkeley.edu/users/aldous/book.html
B. Bercu and A. Rouault, Sharp large deviations for the Ornstein-Uhlenbeck process (to appear).
Bolthausen, E., The Berry-Esseen Theorem for Functionals of Discrete Markov Chains. Z. Wahrscheinlichkeitstheorie Verw. 54 (1980) 59-73. CrossRef
Bryc, W. and Dembo, A., Large deviations for quadratic functionals of gaussian processes. J. Theoret. Probab. 10 (1997) 307-332. CrossRef
Cheng, M.F. and Wang, F.Y., Estimation of spectral gap for elliptic operators. Trans. AMS 349 (1997) 1239-1267. CrossRef
K.L. Chung. Markov chains with stationnary transition probabilities. Springer-Verlag (1960).
J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, Boston (1989).
P. Diaconis, S. Holmes and R.M. Neal, Analysis of a non-reversible markov chain sampler, Technical Report. Cornell University, BU-1385-M, Biometrics Unit (1997).
Dinwoodie, I.H., A probability inequality for the occupation measure of a reversible Markov chain. Ann. Appl. Probab 5 (1995) 37-43. CrossRef
Dinwoodie, I.H., Expectations for nonreversible Markov chains. J. Math. Ann. App. 220 (1998) 585-596. CrossRef
Dinwoodie, I.H. and Occupation, P Ney measures for Markov chains. J. Theoret. Probab. 8 (1995) 679-691. CrossRef
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley & Sons, 2nd Edition (1971).
S. Gallot and D. Hulin and J. Lafontaine, Riemannian Geometry. Springer-Verlag (1990).
D. Gillman, Hidden Markov Chains: Rates of Convergence and the Complexity of Inference, Ph.D. Thesis. Massachusetts Institute of Technology (1993).
L. Gross, Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups, in Dirichlet forms, Varenna (Italy). Springer-Verlag, Lecture Notes in Math. 1563 (1992) 54-88.
J.L. Jensen, Saddlepoint Approximations. Oxford Statist. Sci. Ser. 16 .
T. Kato, Perturbation theory for linear operators. Springer (1966).
Landers, D. and Rogge, L., On the rate of convergence in the central limit theorem for Markov chains. Z. Wahrscheinlichkeitstheorie Verw. 35 (1976) 169-183.
Lawler, G.F. and Sokal, A.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) 557-580.
Lezaud, P., Chernoff-type Bound for Finite Markov Chains. Ann. Appl. Probab 8 (1998) 849-867.
B. Mann, Berry-Esseen Central Limit Theorem for Markov chains, Ph.D. Thesis. Harvard University (1996).
Marton, K., A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (1996) 556-571. CrossRef
Nagaev, S.V., Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 (1957) 378-406. CrossRef
P.M. Samson, Concentration of measure inequalities for Markov chains and Φ-mixing processes, Ann. Probab. 28 (2000) 416-461.
Trotter, H.F., On the product of semi-groups of operators. Proc. Amer. Math. Soc. 10 (1959) 545-551. CrossRef
F.Y. Wang, Existence of spectral gap for elliptic operators. Math. Sci. Res. Inst. (1998).