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Approximate decomposition of some modulated-Poisson Voronoi tessellations

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Bartłomiej Błaszczyszyn  and
René Schott
Bartłomiej Błaszczyszyn*
Affiliation:
ENS/INRIA, Paris, and University of Wrocław
René Schott*
Affiliation:
Université Henri Poincaré-Nancy 1
*
∗ Postal address: ENS, 45 rue d'Ulm, 75230 Paris, France. Email address: bartek.blaszczyszyn@ens.fr
∗∗ Postal address: Université Henri Poincaré-Nancy 1, BP 239, 54506 Vandoeuvre-lès-Nancy, France.
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Abstract

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

Keywords

Voronoi tessellationinhomogeneous Poisson point processdouble stochastic Poisson point processBoolean modelintensity measureapproximationsdecomposability

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 847 - 862
Copyright
Copyright © Applied Probability Trust 2003 

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