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

### On random sets connected to the partial records of Poisson point process

Auteur(s):

Code(s) de Classification MSC:

• 60D05 Geometric probability, stochastic geometry, random sets, See also {52A22, 53C65}

Résumé: Random intervals are constructed from partial records in a Poisson point process in $]0,\infty[\times]0,\infty[.$ These are used to cover partially $[0,\infty[$; the purpose of this work is to study the random set $\Rs$ that is left uncovered. We show that $\Rs$ enjoys the regenerative property and identify its distribution in terms of the characteristics of the Poisson point process. As an application we show that $\Rs$ is almost surely a fractal set and we calculate its dimension.

Mots Clés: Poisson point process ; Extremal Process ; Regenerative sets ; Subordinators ; Fractal dimensions

Date: 2001-06-26

Prépublication numéro: PMA-674