Some Asymptotic Results for the Number of Generalized Records

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Niveau: Supérieur, Doctorat, Bac+8
Some Asymptotic Results for the Number of Generalized Records ANNE-LAURE FOUGERES Laboratoire de Statistique et de Probabilites, Universite Paul Sabatier, 118 Route de Narbonne, F-31062, Toulouse Cedex 04, France E-mail: FABRICE GAMBOA Laboratoire de Statistique et de Probabilites, Universite Paul Sabatier, 118 Route de Narbonne, F-31062, Toulouse Cedex 04, France E-mail: CLEMENTINE PRIEUR Dept GMM, Laboratoire de Statistique et Probabilites, INSA, 135 Avenue de Rangueil, F-31077, Toulouse Cedex 04, France E-mail: [Received 8 April 2004; Revised 7 March 2005; Accepted 18 May 2005] Abstract. In this paper, we study asymptotic properties (large deviations and functional central limit theorem) of generalized record processes built on a triangular array of continuous and exchangeable random variables. As an application of these results, the links with the Kendall's rank correlation statistic are studied and testing exchangeability is discussed. Key words. exchangeable distribution, generalized records, large deviations, rank AMS 2000 Subject Classification. PrimaryV60F10 SecondaryV60F17, 62G10 1. Introduction Let X ? ?X ?n?k ?n2N*;1kn be a triangular array of exchangeable continuous random variables. That is, for any n 2 N*, we assume that the distribution of the random vector X (n) = (Xk (n))k = 1, .

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Extremes 8, 27–41, 2005 # 2005 Springer Science + Business Media, Inc. Manufactured in The United States.
Some Asymptotic Results for the Number of Generalized Records
` ANNE-LAURE FOUGERES LaboratoiredeStatistiqueetdeProbabilite´s,Universit´ePaulSabatier,118RoutedeNarbonne, F-31062, Toulouse Cedex 04, France E-mail: fougeres@insa-toulouse.fr FABRICE GAMBOA LaboratoiredeStatistiqueetdeProbabilit´es,Universit´ePaulSabatier,118RoutedeNarbonne, F-31062, Toulouse Cedex 04, France E-mail: gamboa@math.ups-tlse.fr ´ CLEMENTINE PRIEUR D´eptGMM,LaboratoiredeStatistiqueetProbabilite´s,INSA,135AvenuedeRangueil,F-31077,Toulouse Cedex 04, France E-mail: prieur@cict.fr [Received 8 April 2004; Revised 7 March 2005; Accepted 18 May 2005]
Abstract. In this paper, we study asymptotic properties (large deviations and functional central limit theorem) of generalized record processes built on a triangular array of continuous and exchangeable random variables. As an application of these results, the links with the Kendall’s rank correlation statistic are studied and testing exchangeability is discussed. Key words. exchangeable distribution, generalized records, large deviations, rank Primary V 60F10 2000 ect Classification. AMS Subj Secondary V 60F17, 62G10
1. Introduction Let X ¼ ð X k ð n Þ Þ n 2 N * ; 1 k n be a triangular array of exchangeable continuous random variables. That is, for any n 2 N *, we assume that the distribution of the random vector X ( n ) = ( X k ( n ) ) k = 1, . . . , n is invariant under any permutation of the indexes {1, 2, . . . , n }. In the particular case of a sequence of exchangeable random variables, that is when X j ( n ) = X j for any n 2 N * and j 2 1, 2, . . . , n , de Fi ( ne ) t(ti n ’s 2 th N e*o) : reInmde(eAdldous,1983)givesa representation formula for the distribution of X n , it is well known that in this case the distribution of X ( n ) is that of a mixture of vectors of independent and identically distributed (i.i.d.) variables. In the whole paper we assume that, for n 2 N * 0 and j , j 0 2 {1,  , n }, j m j , X j ( n ) m X j ( 0 n ) (a.s.). Following Resnick (1987) we define, for