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Approximate decomposition of some modulated-Poisson Voronoi tessellations

Bartlomiej Blaszczyszyn 1 René Schott
1 TREC - Theory of networks and communications
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt
Abstract : We consider the Voronoi tessellation of Euclidian 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 to 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 remaining term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decomposition are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of non-homogeneous-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.
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https://hal.inria.fr/inria-00072000
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Submitted on : Tuesday, May 23, 2006 - 7:30:15 PM
Last modification on : Tuesday, September 22, 2020 - 3:51:46 AM
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Bartlomiej Blaszczyszyn, René Schott. Approximate decomposition of some modulated-Poisson Voronoi tessellations. [Research Report] RR-4585, INRIA. 2002. ⟨inria-00072000⟩

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