I. Dans and L. Cas-g-´-enéralen´enéral-condition-de-doeblin, Rappelons que la condition (D 0 ) estéquivalentèestéquivalentè a la propriété de ?-mélange (cf lemme 5-(3.9)) condition plus forte que celle de ?-mélange

H. =. , M. , H. , P. M. , P. De et al., P est donc un opérateur borné défini sur l'espace de Hilbert L 2 (?) et l'existence d'une unique probabilité invariante implique que 1 est une valeur propre de multiplicité 1 (relation entre le spectre de P et de son adjoint) donc que le projecteur propre noté E ? est de rang fini 1. L'espace H = L 2 (?) admet alors la décomposition Le spectre de P M est constitué d'un seul point, donc P M ? I est quasi-nilpotent (i.e. son rayon spectral est nul) La résolvante R(?) de P peut s'´ ecrire comme ? E ? ), o` u R (?) s'appelle la résolvante réduite de P, est isolée. Notons par (P ) la distance de 1 au reste du spectre de P. Supposonségalement Supposonségalement qu

D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs

D. Bakry, L'hypercontractivit?? et son utilisation en th??orie des semigroupes, Lecture Notes in Math, vol.48, issue.n.2, pp.1-114, 1994.
DOI : 10.1007/978-3-642-96208-0

D. Bakry and M. Emery, Diffusions hypercontractives, Lecture Notes in Math, vol.1123, pp.177-206, 1985.
DOI : 10.1007/BFb0075847

URL : http://archive.numdam.org/article/SPS_1985__19__177_0.pdf

H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Birkhäuser, 1985.

G. Bennett, Probability Inequalities for the Sum of Independent Random Variables, Journal of the American Statistical Association, vol.18, issue.297, pp.33-45, 1962.
DOI : 10.1214/aoms/1177730437

E. Bolthausen, The Berry-Esseen theorem for functionals of discrete Markov chains, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.9, issue.1, pp.59-73, 1980.
DOI : 10.1007/BF00535354

R. C. Bradley, Basic Properties of Strong Mixing Conditions, Dependence in Probability and Statistics, a Survey of Recent Results. Birkhäuser, 1986.
DOI : 10.1007/978-1-4615-8162-8_8

M. Carbon, Inégalité de Bernstein pour les processus fortement mélangeants non nécessairement stationnaires, C. R. Acad. Paris, Série A, issue.297, pp.303-306, 1983.

D. Chauveau and J. Diebolt, MCMC Convergence diagnostic via the Central Limit Theorem. preprint, 1997.

K. L. Chung, Markov chains with stationnary transition probabilities, 1960.

K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, 1995.

G. Collomb, Propriétés de convergence presquecompì ete du prédicteurprédicteur`prédicteurà noyau, Z. Wahrsch. Verw. Gebiete, issue.66, pp.441-460, 1984.
DOI : 10.1007/bf00533708

P. Diaconis and L. Saloff-coste, What do we know about the Metropolis algorithm ?, 1995.

I. H. Dinwoodie, A Probability Inequality for the Occupation Measure of a Reversible Markov Chain, The Annals of Applied Probability, vol.5, issue.1, pp.37-43, 1995.
DOI : 10.1214/aoap/1177004826

J. L. Doob, Stochastic Processes, 1953.

P. Doukhan, Mixing : Properties and Examples, 1994.

N. Dunford and J. T. Schwartz, Linear Operators. Part I. General Theory, 1958.

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, 1990.

. P. Stewart, T. G. Ethier, and . Kurtz, Markov processes ; Characterization and convergence, 1986.

W. Feller, An Introduction to Probability Theory and Its Applications, 1971.

W. Feller, An Introduction to Probability Theory and Its Applications, 1971.

O. Forster, Lectures on Riemann Surfaces, volume 81 of Graduate texts in Mathematics, 1993.

D. Gillman, Hidden Markov Chains : Rates of Convergence and the Complexity of Inference, 1993.

B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, 1954.

L. Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups, Dirichlet forms, pp.54-88, 1992.
DOI : 10.1007/BFb0074091

T. E. Harris, The Theory of Branching Processes, 1963.
DOI : 10.1007/978-3-642-51866-9

I. A. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables, 1971.

I. A. Ibragimov, Some Limit Theorems for Stationary Processes, Theory of Probability & Its Applications, vol.7, issue.4, pp.349-382, 1962.
DOI : 10.1137/1107036

I. A. Ibragimov, A Note on the Central Limit Theorems for Dependent Random Variables, Theory of Probability & Its Applications, vol.20, issue.1, pp.135-141, 1975.
DOI : 10.1137/1120011

J. L. Jensen, Saddlepoint Approximations, Oxford Statistical Science, vol.16

T. Kato, Perturbation theory for linear operators, 1966.

K. Knopp, Theory of functions, Kolmogorov. ¨ Uber das Gesetz des iterieten Logaritmus, pp.309-319, 1929.

G. F. Lawler and A. D. Sokal, Bounds on the L 2 spectrum for Markov chains and Markov processes : a generalization of Cheeger's inequality, Trans. Amer. Math. Soc, vol.309, pp.557-580, 1988.

P. Lezaud, Chernoff-type bound for finite Markov chains, The Annals of Applied Probability, vol.8, issue.3, pp.849-867, 1998.
DOI : 10.1214/aoap/1028903453

URL : https://hal.archives-ouvertes.fr/hal-00940907

B. Mann, Berry-Esseen Central Limit Theorem For Markov chains, 1996.

A. W. Marshall and I. Olkin, Inequalities : Theory of Majorization and Its Applications, 1979.
DOI : 10.1007/978-0-387-68276-1

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 1993.

S. V. Nagaev, Some Limit Theorems for Stationary Markov Chains, Theory of Probability & Its Applications, vol.2, issue.4, pp.378-406, 1957.
DOI : 10.1137/1102029

S. V. Nagaev, More Exact Statement of Limit Theorems for Homogeneous Markov Chains, Theory of Probability & Its Applications, vol.6, issue.1, pp.62-81, 1961.
DOI : 10.1137/1106005

J. Neveu, Bases mathématiques du calcul des probabilités. Masson, 2` eme edition, 1980.

V. V. Petrov, Sums of Independent Random Variables, 1975.
DOI : 10.1007/978-3-642-65809-9

J. W. Pitman, Occupation measures for Markov chains, Advances in Applied Probability, vol.3, issue.01, pp.69-86, 1977.
DOI : 10.1090/S0002-9939-1954-0060757-0

E. Seneta, Non-negative Matrices and Markov Chains, 1980.
DOI : 10.1007/0-387-32792-4

W. F. Stout, Almost Sure Convergence, 1974.

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer, pp.545-551, 1959.
DOI : 10.1090/S0002-9939-1959-0108732-6

J. H. Wilkinson, The Algebraic Eigenvalue Problem, 1965.

K. Yosida and S. Kakutani, Operator-Theoretical Treatment of Markoff's Process and Mean Ergodic Theorem, The Annals of Mathematics, vol.42, issue.1, 1941.
DOI : 10.2307/1968993