Convergence of the normalized maximum of regularly varying random functions in the space $ \mathbb{D}$

Yoann Gentric

doi:10.1016/j.crma.2008.01.007

Probability Theory

Convergence of the normalized maximum of regularly varying random functions in the space

D

Presented by: Paul Deheuvels

Yoann Gentric ¹

¹ Laboratoire Modal'X, Université Paris 10, 92000 Nanterre, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 329-334.

Abstract
Résumé

Let $ξ, ξ_{1}, ξ_{2}, \dots$ be i.i.d. random functions in the space $D$ of cadlag functions. The purpose of this note is to complement the result of de Haan and Lin (2001) on the link between regular variation of ξ and convergence of the normalized maximum $n^{−1} ⋁_{i = 1}^{n} ξ_{i}$ in the space $C$ of continuous functions. We study when regular variation implies convergence of the normalized maximum in $D$ . After exhibiting an example, which shows that this is not true in the general case, we give a sufficient condition under which this implication takes place.

Soit $ξ, ξ_{1}, ξ_{2}, \dots$ des fonctions aléatoires i.i.d. dans l'espace $D$ des fonctions cadlag. De Hann et Lin (2001) ont étudié le lien entre la variation régulière de ξ et la convergence en loi dans $C$ du maximum renormalisé $n^{−1} ⋁_{i = 1}^{n} ξ_{i}$ . Après avoir exhibé un contre-exemple qui montre que le résultat est faux en toute généralité dans $D$ , nous donnons une condition suffisante qui assure la convergence du maximum renormalisé dans $D$ . A titre d'exemple, le cas d'un processus de Lévy est traité.

Received: 2006-11-14
Accepted: 2008-01-14
Published online: 2008-02-11

DOI: 10.1016/j.crma.2008.01.007

Author's affiliations:

Yoann Gentric ¹

¹ Laboratoire Modal'X, Université Paris 10, 92000 Nanterre, France

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     title = {Convergence of the normalized maximum of regularly varying random functions in the space $ \mathbb{D}$},
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Yoann Gentric. Convergence of the normalized maximum of regularly varying random functions in the space $ \mathbb{D}$. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 329-334. doi : 10.1016/j.crma.2008.01.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.007/

References
Cited by

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[3] L. de Haan; T. Lin On convergence toward an extreme value distribution in $C [0, 1]$ , Ann. Probab., Volume 29 (2001) no. 1, pp. 467-483

[4] H. Hult; F. Lindskog On regular variation for infinitely divisible random vectors and additive processes, Adv. Appl. Probab., Volume 38 (2006) no. 1, pp. 134-148

[5] S.I. Resnick Extreme Values, Regular Variation, and Point Processes, Applied Probability, A series of the Applied Probability Trust, vol. 4, Springer-Verlag, New York, 1987

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