Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains
ESAIM: Probability and Statistics, Tome 7 (2003), pp. 115-146.

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≥0 k r P k f, r∈ℕ, under some regularity assumptions and implies the central limit theorem with a rate in n -1 2 for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

DOI : 10.1051/ps:2003003
Classification : 60J10, 37A05, 37A25
Mots clés : transfer operator, convergence of iterates, Markov chains, rate in the TCL for dynamical systems, Borel-Cantelli property, non uniformly hyperbolic map 
@article{PS_2003__7__115_0,
     author = {Conze, Jean-Pierre and Raugi, Albert},
     title = {Convergence of iterates of a transfer operator, application to dynamical systems and to {Markov} chains},
     journal = {ESAIM: Probability and Statistics},
     pages = {115--146},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2003},
     doi = {10.1051/ps:2003003},
     mrnumber = {1956075},
     zbl = {1018.60072},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2003003/}
}
TY  - JOUR
AU  - Conze, Jean-Pierre
AU  - Raugi, Albert
TI  - Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains
JO  - ESAIM: Probability and Statistics
PY  - 2003
SP  - 115
EP  - 146
VL  - 7
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2003003/
DO  - 10.1051/ps:2003003
LA  - en
ID  - PS_2003__7__115_0
ER  - 
%0 Journal Article
%A Conze, Jean-Pierre
%A Raugi, Albert
%T Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains
%J ESAIM: Probability and Statistics
%D 2003
%P 115-146
%V 7
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2003003/
%R 10.1051/ps:2003003
%G en
%F PS_2003__7__115_0
Conze, Jean-Pierre; Raugi, Albert. Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 115-146. doi : 10.1051/ps:2003003. http://archive.numdam.org/articles/10.1051/ps:2003003/

[1] V. Baladi, Positive Transfer Operators and Decay of Correlations. World Scientific, Adv. Ser. Nonlinear Dynam. 16 (2000). | MR | Zbl

[2] R. Bowen, Equilibrium states and the ergodic theory of Anosov Diffeomorphisms. Springer-Verlag, Lectures Notes 470 (1975). | MR | Zbl

[3] B.M. Brown, Martingale central limit theorem. Ann. Math. Statist. 42 (1971) 59-66. | MR | Zbl

[4] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures. Isreal J. Math. 122 (2001) 1-27. | MR | Zbl

[5] J.-P. Conze and A. Raugi, Fonctions harmoniques pour un opérateur de transition et applications. Bull. Soc. Math. France 118 (1990) 273-310. | EuDML | Numdam | MR | Zbl

[6] 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. | EuDML | MR | Zbl

[7] M.I. Gordin, On the central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl. 10 (1969) 1174-1176. | MR | Zbl

[8] M.I. Gordin and B.A. Lifvsic, Central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239 (1978) 766-767. | MR | Zbl

[9] S. Gouëzel, Sharp polynomial estimates for the decay of correlations. Preprint (2002). | MR | Zbl

[10] P. Hall and C.C. Heyde, Martingale limit theory and its applications. Academic Press, New York (1980). | MR | Zbl

[11] 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). | MR | Zbl

[12] H. Hu, Decay of correlations for piecwise smooth maps with indifferent fixed points. Preprint. | Zbl

[13] C. Jan, Vitesse de convergence dans le TCL pour certaines chaînes de Markov et certains systèmes dynamiques, Preprint. Université de Rennes 1 (2000). | MR

[14] D.Y. Kleinbock and G.A. Margulis, Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999) 451-494. | MR | Zbl

[15] A. Kondah, V. Maume and B. Schmitt, Vitesse de convergence vers l'état d'équilibre pour des dynamiques markoviennes non höldériennes. Ann. Inst. H. Poincaré 33 (1997) 675-695. | Numdam | Zbl

[16] C. Liverani, Decay of correlations. Ann. Math. 142 (1995) 239-301. | MR | Zbl

[17] C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 (1999) 671-685. | MR | Zbl

[18] W. Philipp, Some metrical theorems in number theory. Pacific J. Math. 20 (1967) 109-127. | MR | Zbl

[19] M. Pollicott, Rates of mixing for potentials of summable variation. Trans. Amer. Math. Soc. 352 (2000) 843-853. | MR | Zbl

[20] M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points. Comm. Math. Phys. 217 (2001) 503-520. | MR | Zbl

[21] A. Raugi, Théorie spectrale d'un opérateur de transition sur un espace métrique compact. Ann. Inst. H. Poincaré 28 (1992) 281-309. | Numdam | Zbl

[22] E. Rio, Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. J. Probab. Theory Related Fields 104 (1996) 255-282. | Zbl

[23] O. Sarig, Subexponential decay of decorrelation. Preprint (2001). | Zbl

[24] Ya.G. Sinai, Gibbs measures in ergodic theory. Russian Math. Surveys 166 (1972) 21-64. | MR | Zbl

[25] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978) 121-153. | MR | Zbl

[26] L.-S. Young, Recurrence times and rates of mixing. Israel J. Math. 110 (1999) 153-188. | MR | Zbl

Cité par Sources :