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Gamma distributions for stationary Poisson flat processes

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Volker Baumstark  and
Günter Last
Volker Baumstark*
Affiliation:
Universität Karlsruhe
Günter Last*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Institut für Stochastik, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany.
∗ Postal address: Institut für Stochastik, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany.
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Abstract

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We consider a stationary Poisson process X of k-flats in ℝd with intensity measure Θ and a measurable set S of k-flats depending on F1,…,Fn∈ X, x∈ℝd, and X in a specific equivariant way. If (F1,…,Fn,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), 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 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 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.

Keywords

Stochastic geometrygamma distributionk-flatPoisson processPalm distributionstopping setVoronoi tessellationPoisson hyperplane tessellationtypical face

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 911 - 939
Copyright
Copyright © Applied Probability Trust 2009 

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