Stopping Sets: Gamma-Type Results and Hitting Properties

Overview & Synopsis

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.