Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains

J. Dedecker; S. Gouëzel; F. Merlevède

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 3, page 796-821
  • ISSN: 0246-0203

Abstract

top
We consider a large class of piecewise expanding maps T of [0, 1] with a neutral fixed point, and their associated Markov chains Yi whose transition kernel is the Perron–Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f○Ti satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Yi) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f○Ti may belong to the domain of normal attraction of a stable law of index p∈(1, 2), we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.

How to cite

top

Dedecker, J., Gouëzel, S., and Merlevède, F.. "Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains." Annales de l'I.H.P. Probabilités et statistiques 46.3 (2010): 796-821. <http://eudml.org/doc/239896>.

@article{Dedecker2010,
abstract = {We consider a large class of piecewise expanding maps T of [0, 1] with a neutral fixed point, and their associated Markov chains Yi whose transition kernel is the Perron–Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f○Ti satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Yi) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f○Ti may belong to the domain of normal attraction of a stable law of index p∈(1, 2), we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.},
author = {Dedecker, J., Gouëzel, S., Merlevède, F.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {intermittency; almost sure convergence; law of the iterated logarithm; strong invariance principle},
language = {eng},
number = {3},
pages = {796-821},
publisher = {Gauthier-Villars},
title = {Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains},
url = {http://eudml.org/doc/239896},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Dedecker, J.
AU - Gouëzel, S.
AU - Merlevède, F.
TI - Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 796
EP - 821
AB - We consider a large class of piecewise expanding maps T of [0, 1] with a neutral fixed point, and their associated Markov chains Yi whose transition kernel is the Perron–Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f○Ti satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Yi) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f○Ti may belong to the domain of normal attraction of a stable law of index p∈(1, 2), we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.
LA - eng
KW - intermittency; almost sure convergence; law of the iterated logarithm; strong invariance principle
UR - http://eudml.org/doc/239896
ER -

References

top
  1. [1] E. Berger. An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables. Probab. Theory Related Fields 84 (1990) 161–201. Zbl0695.60041MR1030726
  2. [2] J. Dedecker and F. Merlevède. Convergence rates in the law of large numbers for Banach-valued dependent variables. Teor. Veroyatn. Primen. 52 (2007) 562–587. Zbl1158.60009MR2743029
  3. [3] J. Dedecker and C. Prieur. Some unbounded functions of intermittent maps for which the central limit theorem holds. ALEA Lat. Am. J. Probab. Math. Stat. 5 (2009) 29–45. Zbl1160.37354MR2475605
  4. [4] J. Dedecker and E. Rio. On mean central limit theorems for stationary sequences. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 693–726. Zbl1187.60015MR2446294
  5. [5] C.-G. Esseen and S. Janson. On moment conditions for normed sums of independent variables and martingale differences. Stochastic Process. Appl. 19 (1985) 173–182. Zbl0554.60050MR780729
  6. [6] W. Feller. An Introduction to Probability Theory and Its Applications. Vol. II. Wiley, New York–London–Sydney, 1966. Zbl0138.10207MR210154
  7. [7] W. Feller. An extension of the law of the iterated logarithm to variables without variance. J. Math. Mech. 18 (1968) 343–355. Zbl0254.60016MR233399
  8. [8] M. I. Gordin. Abstracts of communication, T.1:A-K. In International Conference on Probability Theory, Vilnius, 1973. 
  9. [9] S. Gouëzel. Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 (2004) 82–122. Zbl1038.37007MR2027296
  10. [10] S. Gouëzel. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004) 29–65. Zbl1070.37003MR2041223
  11. [11] S. Gouëzel. Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes. Thèse 7526, l’Université Paris Sud, 2004. 
  12. [12] S. Gouëzel. A Borel–Cantelli lemma for intermittent interval maps. Nonlinearity 20 (2007) 1491–1497. Zbl1120.34004MR2327135
  13. [13] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin, 2001. Zbl0983.60005MR1862393
  14. [14] C. C. Heyde. A note concerning behaviour of iterated logarithm type. Proc. Amer. Math. Soc. 23 (1969) 85–90. Zbl0185.46901MR251772
  15. [15] F. Hofbauer and G. Keller. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982) 119–140. Zbl0485.28016MR656227
  16. [16] C. Liverani, B. Saussol and S. Vaienti. A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 (1999) 671–685. Zbl0988.37035MR1695915
  17. [17] I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260 (2005) 131–146. Zbl1084.37024MR2175992
  18. [18] F. Merlevède. On a maximal inequality for strongly mixing random variables in Hilbert spaces. Application to the compact law of the iterated logarithm. Publ. Inst. Stat. Univ. Paris 12 (2008) 47–60. MR2435040
  19. [19] W. Philipp and W. F. Stout. Almost Sure Invariance Principle for Partial Sums of Weakly Dependent Random Variables. Mem. Amer. Math. Soc. 161. Amer. Math. Soc., Providence, RI, 1975. Zbl0361.60007
  20. [20] I. Pinelis. Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22 (1994) 1679–1706. Zbl0836.60015MR1331198
  21. [21] Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74 (1980) 189–197. Zbl0578.76059MR576270
  22. [22] E. Rio. The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23 (1995) 1188–1203. Zbl0833.60024MR1349167
  23. [23] E. Rio. Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications de la SMAI 31. Springer, Berlin, 2000. Zbl0944.60008MR2117923
  24. [24] M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43–47. Zbl0070.13804MR74711
  25. [25] O. Sarig. Subexponential decay of correlations. Invent. Math. 150 (2002) 629–653. Zbl1042.37005MR1946554
  26. [26] W. F. Stout. Almost Sure Convergence. Academic Press, New-York, 1974. Zbl0321.60022MR455094
  27. [27] D. Volný and P. Samek. On the invariance principle and the law of iterated logarithm for stationary processes. In Mathematical Physics and Stochastic Analysis 424–438. World Scientific., River Edge, 2000. Zbl0974.60013MR1893125
  28. [28] L.-S. Young. Recurrence times and rates of mixing. Israel J. Math. 110 (1999) 153–188. Zbl0983.37005MR1750438
  29. [29] R. Zweimüller. Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 (1998) 1263–1276. Zbl0922.58039MR1644385

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.