Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On the ranked excursions heights of a Kiefer process

Auteur(s):

Code(s) de Classification MSC:

• 60F15 Strong theorems
• 60G55 Point processes

Résumé: Let $(K(s,t), 0\le s\le 1, \, t\ge1)$ be a Kiefer process, i.e. a continuous two-parameter centered Gaussian process indexed by $[0,1]\times \r_+$ whose covariance function is given by $\e\big( K(s_1,t_1) K(s_2, t_2)\big) = (\min(s_1, s_2)- s_1 s_2) \, \min( t_1 , t_2), \, 0\le s_1, s_2 \le 1, \, t_1, t_2\ge0.$ For each $t>0$, the process $K(\cdot, t)$ is a Brownian bridge on the scale of $\sqrt t$. Let $M^*_1(t) \ge M^*_2(t) \ge... M^*_j(t)\ge...0$ be the ranked excursions heights of $K(\cdot, t)$. In this paper, we study the path properties of the process $t\to M^*_j(t)$. Two laws of iterated logarithm are established to describe the asymptotic behaviors of $M^*_j(t)$ as $t$ goes to infinity.

Mots Clés: Kiefer process ; excursions ; ranked heights

Date: 2002-06-27

Prépublication numéro: PMA-743

Pdf file : PMA-743.pdf