# Stopping Sets: Gamma-Type Results and Hitting Properties

Abstract : Recently in the paper \cite{MolZuy:96} there was established the following Gamma-type result. Given the number $N$ of a homogeneous Poisson process' points defining a random figure, its volume is $\Gamma(N,\lambda)$ distributed, where $\lambda$ is the intensity of the process. The goal of this paper is to give an alternative description of the class of the random sets for which the Gamma-type results hold. We show that it corresponds to the class of \emph{stopping sets} with respect to the natural filtration of the point process with certain scaling properties. The proof is very short and uses the martingale technique for directed processes, in particular, the analog of the Doob's optional sampling theorem proved in \cite{Kur:80}. Along with an elegance, this approach provides a new inside into the nature of geometrical objects constructed with respect to a point process. We show, in particular, that in the Poisson case the probability of a point to be covered by a stopping set does not depend on whether it is point of the Poisson process or not.
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Type de document :
Rapport
RR-3146, INRIA. 1997
Domaine :

https://hal.inria.fr/inria-00073543
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 13:09:53
Dernière modification le : samedi 27 janvier 2018 - 01:31:30
Document(s) archivé(s) le : dimanche 4 avril 2010 - 22:00:51

### Identifiants

• HAL Id : inria-00073543, version 1

### Citation

Sergei Zuyev. Stopping Sets: Gamma-Type Results and Hitting Properties. RR-3146, INRIA. 1997. 〈inria-00073543〉

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