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Convergence of iterates of a transfer operator,application to dynamical systems and to Markov chains

Published online by Cambridge University Press:  15 May 2003

Jean-Pierre Conze  and
Albert Raugi
Jean-Pierre Conze
Affiliation:
IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France; conze@univ-rennes1.fr.
Albert Raugi
Affiliation:
IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France; conze@univ-rennes1.fr.
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Abstract

We present a spectral theory for a class of operators satisfying a weak“Doeblin–Fortet" condition and apply it to a class of transition operators.This gives the convergence of the series ∑k≥0krPkƒ, $r \in \mathbb{N}$ ,under some regularity assumptions and implies the central limit theoremwith a rate in $n^{- \frac{1}{2} }$ for the corresponding Markov chain.An application to a non uniformly hyperbolic transformation on the interval is also given.

Keywords

Transfer operatorconvergence of iteratesMarkov chainsrate in the TCLfor dynamical systemsBorel-Cantelli propertynon uniformly hyperbolic map.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 115 - 146
Copyright
© EDP Sciences, SMAI, 2003

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References

V. Baladi, Positive Transfer Operators and Decay of Correlations. World Scientific, Adv. Ser. Nonlinear Dynam. 16 (2000).
R. Bowen, Equilibrium states and the ergodic theory of Anosov Diffeomorphisms. Springer-Verlag, Lectures Notes 470 (1975).
Brown, B.M., Martingale central limit theorem. Ann. Math. Statist. 42 (1971) 59-66. CrossRef
Chernov, N. and Kleinbock, D., Dynamical Borel-Cantelli lemmas for Gibbs measures. Isreal J. Math. 122 (2001) 1-27. CrossRef
Conze, J.-P. and Raugi, A., Fonctions harmoniques pour un opérateur de transition et applications. Bull. Soc. Math. France 118 (1990) 273-310. CrossRef
J.-P. Conze and A. Raugi, Convergence des potentiels pour un opérateur de transfert, applications aux systèmes dynamiques et aux chaînes de Markov. Séminaires de Rennes (1998) 52.
M.I. Gordin, On the central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl. 10 (1969) 1174-1176.
Gordin, M.I. and Lifvsic, B.A., Central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239 (1978) 766-767.
S. Gouëzel, Sharp polynomial estimates for the decay of correlations. Preprint (2002).
P. Hall and C.C. Heyde, Martingale limit theory and its applications. Academic Press, New York (1980).
H. Hennion and L. Hervé, Limit theorem for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-compactness. Springer-Verlag, Lectures Notes 1766 (2001).
H. Hu, Decay of correlations for piecwise smooth maps with indifferent fixed points. Preprint.
C. Jan, Vitesse de convergence dans le TCL pour certaines chaînes de Markov et certains systèmes dynamiques, Preprint. Université de Rennes 1 (2000).
Kleinbock, D.Y. and Margulis, G.A., Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999) 451-494. CrossRef
Kondah, A., Maume, V. and Schmitt, B., Vitesse de convergence vers l'état d'équilibre pour des dynamiques markoviennes non höldériennes. Ann. Inst. H. Poincaré 33 (1997) 675-695. CrossRef
Liverani, C., Decay of correlations. Ann. Math. 142 (1995) 239-301. CrossRef
Liverani, C., Saussol, B. and Vaienti, S., A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 (1999) 671-685. CrossRef
Philipp, W., Some metrical theorems in number theory. Pacific J. Math. 20 (1967) 109-127. CrossRef
Pollicott, M., Rates of mixing for potentials of summable variation. Trans. Amer. Math. Soc. 352 (2000) 843-853. CrossRef
Pollicott, M. and Yuri, M., Statistical properties of maps with indifferent periodic points. Comm. Math. Phys. 217 (2001) 503-520. CrossRef
Raugi, A., Théorie spectrale d'un opérateur de transition sur un espace métrique compact. Ann. Inst. H. Poincaré 28 (1992) 281-309.
Rio, E., Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. J. Probab. Theory Related Fields 104 (1996) 255-282. CrossRef
O. Sarig, Subexponential decay of decorrelation. Preprint (2001).
Ya.G. Sinai, Gibbs measures in ergodic theory. Russian Math. Surveys 166 (1972) 21-64.
Walters, P., Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978) 121-153. CrossRef
Young, L.-S., Recurrence times and rates of mixing. Israel J. Math. 110 (1999) 153-188. CrossRef