Efron type identities for stopping sets and Poisson hulls

Abstract

We consider a Poisson process $\eta$ on a general space with intensity measure $\lambda$ and a stopping set $Z$ depending on $\eta$. Using in particular 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. Our results apply to general Poisson hulls and predominantly even to more general random sets which are neither assumed to be bounded nor to be stopping sets.