no. 5-6
Probability Theory
Convergence of the normalized maximum of regularly varying random functions in the space D
[Convergence du maximum renormalisé de fonctions aléatoires dans l'espace D]

Présenté par : Paul Deheuvels

Gentric, Yoann1
1 Laboratoire Modal'X, Université Paris 10, 92000 Nanterre, France
Comptes Rendus. Mathématique, Tome 346 (2008) no. 5-6, pp. 329-334.

Soit ξ,ξ1,ξ2,… 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=1nξi. 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.

Let ξ,ξ1,ξ2,… 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=1nξi 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.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2008.01.007
Gentric, Yoann 1

1 Laboratoire Modal'X, Université Paris 10, 92000 Nanterre, France 
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Gentric, Yoann. Convergence of the normalized maximum of regularly varying random functions in the space $ \mathbb{D}$. Comptes Rendus. Mathématique, Tome 346 (2008) no. 5-6, pp. 329-334. doi : 10.1016/j.crma.2008.01.007. http://www.numdam.org/articles/10.1016/j.crma.2008.01.007/

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

[5] Resnick, S.I. 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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