Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T02:36:39.419Z Has data issue: false hasContentIssue false
  • English
  • Français

The empirical distribution function for dependent variables:asymptotic and nonasymptotic results in ${\mathbb L}^p$

Published online by Cambridge University Press:  31 March 2007

Jérôme Dedecker  and
Florence Merlevède
Jérôme Dedecker
Affiliation:
Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France; [email protected]
Florence Merlevède
Affiliation:
Laboratoire de probabilités et modèles aléatoires, UMR 7599, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France; [email protected]
Get access

Abstract

Considering the centered empirical distribution function F n-F asa variable in ${\mathbb L}^p(\mu)$ , we derive non asymptotic upperbounds for the deviation of the ${\mathbb L}^p(\mu)$ -norms ofF n-F as well as central limit theorems for the empirical processindexed by the elements of generalized Sobolev balls. These resultsare valid for a large class of dependent sequences, includingnon-mixing processes and some dynamical systems.

Keywords

Deviation inequalitiesweak dependenceCramér-von Mises statisticsempirical processexpanding maps.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 102 - 114
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Azuma, K., Weighted sums of certain dependent random variables. Tôkohu Math. J. 19 (1967) 357–367.
H.C.P. Berbee, Random walks with stationary increments and renewal theory. Mathematical Centre Tracts 112, Mathematisch Centrum, Amsterdam (1979).
Birgé, L. and Massart, P., An adaptive compression algorithm in Besov Spaces. Constr. Approx. 16 (2000) 1–36. CrossRef
Collet, P., Martinez, S. and Schmitt, B., Exponential inequalities for dynamical measures of expanding maps of the interval. Probab. Theory Relat. Fields 123 (2002) 301–322. CrossRef
Dedecker, J. and Merlevède, F., The conditional central limit theorem in Hilbert spaces. Stoch. Processes Appl. 108 (2003) 229–262. CrossRef
Dedecker, J. and Prieur, C., Coupling for $\tau$ -dependent sequences and applications. J. Theoret. Probab. 17 (2004) 861–885. CrossRef
Dedecker, J. and Prieur, C., New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203–236. CrossRef
Dedecker, J. and Rio, E., On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1–34. CrossRef
Doukhan, P., Massart, P. and Rio, E., Invariance principle for absolutely regular empirical processes. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995) 393–427.
Gordin, M.I., The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739–741.
Massart, P., The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269–1283. CrossRef
F. Merlevède and M. Peligrad, On the coupling of dependent random variables and applications, in Empirical process techniques for dependent data, Birkhäuser (2002) 171–193.
P. Oliveira and C. Suquet, ${\mathbb L}^2([0,1])$ weak convergence of the empirical process for dependent variables, in Wavelets and statistics (Villard de Lans 1994), Lect. Notes Statist. 103 (1995) 331–344.
Oliveira, P. and Suquet, C., Weak convergence in ${\mathbb L}^p([0,1])$ of the uniform empirical process under dependence. Statist. Probab. Lett. 39 (1998) 363–370. CrossRef
I.F. Pinelis, An approach to inequalities for the distributions of infinite-dimensional martingales, in Probability in Banach spaces, Proc. Eight Internat. Conf. 8 (1992) 128–134.
Rio, E., Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. C. R. Acad. Sci. Paris Série I 330 (2000) 905–908. CrossRef
van der Vaart, A.W., Bracketing smooth functions. Stoch. Processes Appl. 52 (1994) 93–105. CrossRef
Woyczyński, W.A., A central limit theorem for martingales in Banach spaces. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 23 (1975) 917–920.
Yurinskii, V.V., Exponential bounds for large deviations. Theory Prob. Appl. 19 (1974) 154–155. CrossRef