Gamma distributions for stationary Poisson flat processes

Abstract

We consider a stationary Poisson process $X$ of k-flats in $\R^d$ with intensity measure $\Theta$ and a measurable set $S$ of k-flats depending on F₁,...,F_n∈ X, x∈ℝ^d, and X in a specific equivariant way. If (F₁,...,F_n,x) is properly sampled (in a `typical way') then Θ(S) has a gamma distribution. This result generalizes and unifies earlier work by Miles (1971), M{\o}ller and Zuyev (1996), and Zuyev (1999). As a new example, we will show that the volume of the fundamental region of a typical j-face of a stationary Poisson--Voronoi tessellation is conditionally gamma distributed. This is true in the area-biased and the area-debiased cases. In the first case the shape parameter is not integer valued. As another new example, we will show that the generalized j-face of a Poisson hyperplane tessellation are conditionally gamma distributed. In the isotropic case the contents boil down to the mean breadth of the face.