Gamma distributions for stationary Poisson flat processes

Abstract

We consider a stationary Poisson process X of k-flats in R d with intensity measure and a measurable set S of k-flats depending on F 1 , . . . , F n ∈ X, x ∈ R d , and X in a specific equivariant way. If (F 1 , . . . , F n , x) is properly sampled (in a 'typical way') then (S) has a gamma distribution. This result generalizes and unifies earlier work by Miles (1971), Møller 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 PoissonVoronoi 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 integral-geometric contents of the (area-biased and area-debiased) typical 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.

Weighted faces of Poisson hyperplane tessellations

“…Hence, for each L ∈ G(d, k), the projection X |L is a ball, and equality holds on the right-hand side of (19). Thus, equality holds on the right-hand side of (1). This is also true if the associated zonoid X is an ellipsoid.…”

“…This is also true if the associated zonoid X is an ellipsoid. We have not been able to decide whether this is the only equality case for the right-hand side of (1). Now suppose that X is a parallel mosaic.…”

Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors

“…Therefore, the Minkowski functionals are more robust structure characteristics, which are also suitable for an analysis of noisy data sets. The above described phenomenon extends to typical faces of lower dimensions [10]. For other intrinsic volumes, there seems to be no mathematical argument supporting the occurrence of mixed Gamma distributions.…”

“…Even for Poisson point processes on very general spaces the intensity measure of certain random sets is conditionally Gamma-distributed [108]. A systematic and unifying explanation of these phenomena in a Euclidean setting was given in [10]; see also [9]. One of the results in [73,10] ascertains (for n = 3) that the distribution of the (n − 2)-nd Minkowski functional of the typical cell of a statistically isotropic Poisson hyperplane tessellation is conditionally Gamma-distributed given the number m of neighbors.…”

Intersection and proximity of processes of flats

“…This fact allows us to associate with X a dual Poisson process X ⊥ of (n − k)-flats, which has the same intensity γ and whose directional distribution Q ⊥ is given by the image of Q under the orthogonal complement map L → L ⊥ . Then, writing κ n−2k for the volume of the (n − 2k)-dimensional unit ball and π(X) for π(X, 1), we have the relation π(X) = κ n−2k γ (2)…”