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Comptes Rendus Mathématique
Volume 351, n° 13-14
pages 575-578 (juillet 2013)
Doi : 10.1016/j.crma.2013.07.011
Received : 5 July 2013 ;  accepted : 11 July 2013
Comportement asymptotique de lʼestimateur non paramétrique de la fonction de renouvellement associée à des variables aléatoires positives indépendantes et non stationnaires
Asymptotic behavior of the nonparametric estimator for the renewal function associated with independent and nonstationary positive random variables
 

Michel Harel a, b , Fy Mamenosoa Ravelomanantsoa c
a IUFM du Limousin, 209, bd de Vanteaux, 87036 Limoges cedex, France 
b IMT (UMR CNRS 5219), université Paul-Sabatier, 31062 Toulouse cedex, France 
c Université dʼAntananarivo, faculté des sciences, département de mathématiques et informatique, BP 906, 101 Antananarivo, Madagascar 

Résumé

Nous estimons la fonction de renouvellement, associée à des variables aléatoires positives indépendantes et non stationnaires, par une somme de U -statistiques ou de V -statistiques. Nous étudions ses propriétés asymptotiques : normalité asymptotique, convergence presque sûre ainsi que sa convergence faible. Cette dernière sera traitée dans lʼespace des fonctions définies sur   continues à droite et admettant une limite à gauche, muni de la topologie de Skorokhod.

The full text of this article is available in PDF format.
Abstract

We estimate the renewal function, for independent and nonstationary positive random variables, by considering a sum of U -statistics or V -statistics. We study its asymptotic properties: asymptotic normality, almost sure convergence and weak convergence. This last property will be established on the space of functions defined on   that are right-continuous and have left-hand limits, provided with the Skorokhod topology.

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