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Some distributional results for Poisson-Voronoi tessellations

Part of: Stochastic processes Geometric probability and stochastic geometry

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 the Voronoi tessellation based on a stationary Poisson process N in ℝd. We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d−k+1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

Keywords

Voronoi tessellationPoisson processrandom measurePalm distributiontypical facetypical edge

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 16 - 40
Copyright
Copyright © Applied Probability Trust 2007 

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