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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; dedecker@ccr.jussieu.fr
Florence Merlevède
Affiliation:
Laboratoire de probabilités et modèles aléatoires, UMR 7599, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France; merleve@ccr.jussieu.fr
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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.

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