Stopping sets: Gamma-type results and hitting properties

Recently in the paper by Moller and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is Gamma(n, lambda) distributed, where lambda is the intensity of the process. In this paper we give an alternative description of the class of random sets for which the Gamma-type results hold. We show that it corresponds to the class of stopping sets with respect to the natural filtration of the point process with certain scaling properties. The proof uses the martingale technique for directed processes, in particular, an analogue of Doob's optional sampling theorem proved in Kurtz (1980). As well as being compact, this approach provides a new insight into the nature of geometrical objects constructed with respect to a Poisson point process. We show, in particular, that in this framework the probability that a point is covered by a stopping set does not depend on whether it is a point of the process or not.