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Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Kasra Alishahi  and
Mohsen Sharifitabar
Kasra Alishahi*
Affiliation:
Sharif University of Technology
Mohsen Sharifitabar*
Affiliation:
Sharif University of Technology
*
∗ Postal address: Department of Mathematics, Sharif University of Technology, Tehran, Iran.
∗ Postal address: Department of Mathematics, Sharif University of Technology, Tehran, Iran.
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Abstract

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This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, …. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ−1(1 − e−λu). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

Keywords

Poisson-Voronoi tessellationtypical cellBlaschke-Petkantschin formulahigh dimensionchord length distributionlinear contact distributiongeometric covariogram

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F25: $L^p$-limit theorems 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 919 - 938
Copyright
Copyright © Applied Probability Trust 2008 

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