Efron type identities for stopping sets and Poisson hulls

Abstract

We consider a Poisson process $\eta$ on a general Borel space with intensity measure $\lambda$ and a stopping set $Z$ depending on $\eta$. Generalizing earlier results we prove a spatial Markov property for $Z$. In fact we establish this property for the larger class of finitely insertion tolerant random sets, which include pointwise limits of stopping sets. Using the spatial Markov property of $\eta$, we derive several distributional identities for the restrictions of $\eta$ and $\lambda$ to $Z$ and the complement of $Z$. An important special case in Euclidean space is the convex hull of a finite Poisson process. In this case our results generalize classical (and also more recent) identities connecting the number of vertices and the volume of the convex hull.

35 pages. Major revision and extension