ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, March 2007, Vol. 11, pp. 102-114
DOI: 10.1051/ps:2007009

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

Jérôme Dedecker¹ and Florence Merlevède²

¹ Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France; dedecker@ccr.jussieu.fr
² Laboratoire de probabilités et modèles aléatoires, UMR 7599, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France; merleve@ccr.jussieu.fr

(Received September 20, 2005. Revised May 22, 2006. Published online 31 March 2007.)

Abstract
Considering the centered empirical distribution function F_n-F as a variable in ${\mathbb L}^p(\mu)$ , we derive non asymptotic upper bounds for the deviation of the ${\mathbb L}^p(\mu)$ -norms of F_n-F as well as central limit theorems for the empirical process indexed by the elements of generalized Sobolev balls. These results are valid for a large class of dependent sequences, including non-mixing processes and some dynamical systems.

Mathematics Subject Classification. 60F10, 62G30.

Key words: Deviation inequalities, weak dependence, Cramér-von Mises statistics, empirical process, expanding maps.

© EDP Sciences, SMAI 2007