Abstract
We prove an empirical central limit theorem for the distribution function of a stationary sequence, under a dependence condition involving only indicators of half line. We show that the result applies to the empirical distribution function of iterates of expanding maps with a neutral fixed point at zero as soon as the correlations are summable.
References
Billingsley P.: Convergence of Probability Measures. Wiley, New-York (1968)
Borovkova S., Burton R., Dehling H.: Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Trans. Am. Math. Soc. 353, 4261–4318 (2001)
Collet P., Martinez S., Schmitt B.: Asymptotic distribution of tests for expanding maps of the interval. Ergodic Theory Dynam. Syst. 24, 707–722 (2004)
Dedecker, J., Gouëzel, S., Merlevède, F.: Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains (2008). http://www.lsta.upmc.fr/Dedecker/r7.pdf
Dedecker J., Prieur C.: An empirical central limit theorem for dependent sequences. Stoch. Process. Appl. 117, 121–142 (2007)
Dedecker J., Prieur C.: Some unbounded functions of intermittent maps for which the central limit theorem holds. Alea 5, 29–45 (2009)
Dehling H., Taqqu M.S.: The empirical process of some long-range dependent sequences with an application to U-statistics. Ann. Stat. 17, 1767–1783 (1989)
Dehling, H., Durieu, O., Volný, D.: New techniques for empirical process of dependent data. arXiv:0806.2941v1 (2007)
Doukhan P., Surgailis D.: Functional central limit theorem for the empirical process of short memory linear processes. C. R. Acad. Sci. Paris 326, 87–92 (1998)
Gordin, M.I.: Abstracts of communication, T.1:A-K, International conference on probability theory. Vilnius (1973)
Gouëzel S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory Relat. Fields 128, 82–122 (2004)
Gouëzel S.: A Borel-Cantelli lemma for intermittent interval maps. Nonlinearity 20, 1491–1497 (2007)
Hennion, H., Hervé, L.: Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness. Lecture Notes in Mathematics, vol. 1766. Springer, Heidelberg (2001)
Hofbauer F., Keller G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982)
Liverani C., Saussol B., Vaienti S.: A probabilistic approach to intermittency. Ergodic Theory Dyn. Syst. 19, 671–685 (1999)
Rio E.: Processus empiriques absoluments réguliers et entropie universelle. Probab. Theory Relat. Fields 111, 585–608 (1998)
Rio, E.: Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications de la SMAI, vol. 31. Springer, Heidelberg (2000)
Rosenblatt M.: A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42, 43–47 (1956)
Rozanov Y.A., Volkonskii V.A.: Some limit theorems for random functions I. Theory Probab. Appl. 4, 178–187 (1959)
Vaart A.W., Wellner J.A.: Weak convergence and empirical processes. Springer, Heidelberg (1996)
Wu W.B.: Empirical processes of stationary sequences. Stat. Sin. 18, 313–333 (2008)
Young L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/s00440-011-0393-0.
Rights and permissions
About this article
Cite this article
Dedecker, J. An empirical central limit theorem for intermittent maps. Probab. Theory Relat. Fields 148, 177–195 (2010). https://doi.org/10.1007/s00440-009-0227-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0227-5
Keywords
- Intermittency
- Empirical distribution function
- Rosenthal inequalities
Mathematics Subject Classification (2000)
- 37E05
- 60F17
- 60G10