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

Jean-Pierre Conze; Albert Raugi

ESAIM: Probability and Statistics (2003)

  • Volume: 7, page 115-146
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Conze, Jean-Pierre, and Raugi, Albert. "Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains." ESAIM: Probability and Statistics 7 (2003): 115-146. <http://eudml.org/doc/245526>.

@article{Conze2003,
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 $\sum _\{k \ge 0\} k^r P^k f$, $r \in \mathbb \{N\}$, under some regularity assumptions and implies the central limit theorem with 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.},
author = {Conze, Jean-Pierre, Raugi, Albert},
journal = {ESAIM: Probability and Statistics},
keywords = {transfer operator; convergence of iterates; Markov chains; rate in the TCL for dynamical systems; Borel-Cantelli property; non uniformly hyperbolic map},
language = {eng},
pages = {115-146},
publisher = {EDP-Sciences},
title = {Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains},
url = {http://eudml.org/doc/245526},
volume = {7},
year = {2003},
}

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
PB - EDP-Sciences
VL - 7
SP - 115
EP - 146
AB - 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 $\sum _{k \ge 0} k^r P^k f$, $r \in \mathbb {N}$, under some regularity assumptions and implies the central limit theorem with 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.
LA - eng
KW - transfer operator; convergence of iterates; Markov chains; rate in the TCL for dynamical systems; Borel-Cantelli property; non uniformly hyperbolic map
UR - http://eudml.org/doc/245526
ER -

References

top
  1. [1] V. Baladi, Positive Transfer Operators and Decay of Correlations. World Scientific, Adv. Ser. Nonlinear Dynam. 16 (2000). Zbl1012.37015MR1793194
  2. [2] R. Bowen, Equilibrium states and the ergodic theory of Anosov Diffeomorphisms. Springer-Verlag, Lectures Notes 470 (1975). Zbl0308.28010MR442989
  3. [3] B.M. Brown, Martingale central limit theorem. Ann. Math. Statist. 42 (1971) 59-66. Zbl0218.60048MR290428
  4. [4] N. Chernov and D. Kleinbock, Dynamical Borel–Cantelli lemmas for Gibbs measures. Isreal J. Math. 122 (2001) 1-27. Zbl0997.37002
  5. [5] J.-P. Conze and A. Raugi, Fonctions harmoniques pour un opérateur de transition et applications. Bull. Soc. Math. France 118 (1990) 273-310. Zbl0725.60026MR1078079
  6. [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. Zbl0983.60071MR1794947
  7. [7] M.I. Gordin, On the central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl. 10 (1969) 1174-1176. Zbl0212.50005MR251785
  8. [8] M.I. Gordin and B.A. Lifvsic, Central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239 (1978) 766-767. Zbl0395.60057MR501277
  9. [9] S. Gouëzel, Sharp polynomial estimates for the decay of correlations. Preprint (2002). Zbl1070.37003MR2041223
  10. [10] P. Hall and C.C. Heyde, Martingale limit theory and its applications. Academic Press, New York (1980). Zbl0462.60045MR624435
  11. [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). Zbl0983.60005MR1862393
  12. [12] H. Hu, Decay of correlations for piecwise smooth maps with indifferent fixed points. Preprint. Zbl1071.37026
  13. [13] C. Jan, Vitesse de convergence dans le TCL pour certaines chaînes de Markov et certains systèmes dynamiques, Preprint. Université de Rennes 1 (2000). MR1784921
  14. [14] D.Y. Kleinbock and G.A. Margulis, Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999) 451-494. Zbl0934.22016MR1719827
  15. [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. Zbl0913.60046
  16. [16] C. Liverani, Decay of correlations. Ann. Math. 142 (1995) 239-301. Zbl0871.58059MR1343323
  17. [17] C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 (1999) 671-685. Zbl0988.37035MR1695915
  18. [18] W. Philipp, Some metrical theorems in number theory. Pacific J. Math. 20 (1967) 109-127. Zbl0144.04201MR205930
  19. [19] M. Pollicott, Rates of mixing for potentials of summable variation. Trans. Amer. Math. Soc. 352 (2000) 843-853. Zbl0986.37005MR1621698
  20. [20] M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points. Comm. Math. Phys. 217 (2001) 503-520. Zbl0996.37030MR1822105
  21. [21] A. Raugi, Théorie spectrale d’un opérateur de transition sur un espace métrique compact. Ann. Inst. H. Poincaré 28 (1992) 281-309. Zbl0752.60054
  22. [22] E. Rio, Sur le théorème de Berry–Esseen pour les suites faiblement dépendantes. J. Probab. Theory Related Fields 104 (1996) 255-282. Zbl0838.60017
  23. [23] O. Sarig, Subexponential decay of decorrelation. Preprint (2001). Zbl1042.37005
  24. [24] Ya.G. Sinai, Gibbs measures in ergodic theory. Russian Math. Surveys 166 (1972) 21-64. Zbl0246.28008MR399421
  25. [25] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978) 121-153. Zbl0375.28009MR466493
  26. [26] L.-S. Young, Recurrence times and rates of mixing. Israel J. Math. 110 (1999) 153-188. Zbl0983.37005MR1750438

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.