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Bivariate extension of the Pickands–Balkema–de Haan theorem

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  • Mathematics


doi:10.1016/j.anihpb.2003.03.002 Ann. I. H. Poincaré – PR 40 (2004) 33–41 Bivariate extension of the Pickands–Balkema–de Haan theorem Mario V. Wüthrich Winterthur Insurance, Römerstrasse 17, P.O. Box 357, CH-8401 Winterthur, Switzerland Received 16 September 2002; accepted 25 March 2003 Abstract We prove a two-dimensional version of the famous Pickands–Balkema–de Haan theorem of extreme value theory. The bivariate random variables are generated using the copula language. This representation of dependence structures allows to derive asymptotic results for bivariate excess distributions.  2003 Elsevier SAS. All rights reserved. Résumé Une version en dimension 2 du célèbre théorème de Pickands–Balkema–de Haan sur la théorie des valeurs extrêmes est démontrée. Les variables aléatoires bivariées sont générées en utilisant le langage des copules. Cette représentation des structures de dépendance permet de dériver des résultats asymptotiques pour les distributions d’excès bivariées.  2003 Elsevier SAS. All rights reserved. MSC: 62E20; 62H20; 62P05 Keywords: Archimedean copula; Dependent random variables; Extreme value theory; Pickands–Balkema–de Haan theorem 1. Introduction Copulas were originally introduced about 40 years ago in the context of probabilistic metric spaces. During the past years they have developped rapidly and they have attracted much interest (see, e.g., Kotz–Nadarajah [10]). Copulas are used to describe scale invariant dependencies between random variables. An understanding of such stochastic dependence structures has become very important in all fields of probability theory. Especially in the actuarial world, copulas have proven their usefulness for constructing appropriate multivariate models. An introduction and overviews over recent developments and applications can be found in Joe [7], Nelsen [11], Frees and Valdez [4], Wüthrich [16], Embrechts, McNeil and Straumann [3] and the references therein. In general it is quite dif

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