Open Access
December 2018 Cluster size distributions of extreme values for the Poisson–Voronoi tessellation
Nicolas Chenavier, Christian Y. Robert
Ann. Appl. Probab. 28(6): 3291-3323 (December 2018). DOI: 10.1214/17-AAP1345


We consider the Voronoi tessellation based on a homogeneous Poisson point process in an Euclidean space. For a geometric characteristic of the cells (e.g., the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson–Delaunay tessellation is also discussed.


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Nicolas Chenavier. Christian Y. Robert. "Cluster size distributions of extreme values for the Poisson–Voronoi tessellation." Ann. Appl. Probab. 28 (6) 3291 - 3323, December 2018.


Received: 1 July 2016; Revised: 1 May 2017; Published: December 2018
First available in Project Euclid: 8 October 2018

zbMATH: 06994394
MathSciNet: MR3861814
Digital Object Identifier: 10.1214/17-AAP1345

Primary: 60D05 , 60G70 , 62G32
Secondary: 60F05

Keywords: exceedance point processes , Extreme values , Voronoi tessellations

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 6 • December 2018
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