The Annals of Applied Probability

Cluster size distributions of extreme values for the Poisson–Voronoi tessellation

Nicolas Chenavier and Christian Y. Robert

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 6 (2018), 3291-3323.

Received: July 2016
Revised: May 2017
First available in Project Euclid: 8 October 2018

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 62G32: Statistics of extreme values; tail inference 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems

Extreme values Voronoi tessellations exceedance point processes


Chenavier, Nicolas; Robert, Christian Y. Cluster size distributions of extreme values for the Poisson–Voronoi tessellation. Ann. Appl. Probab. 28 (2018), no. 6, 3291--3323. doi:10.1214/17-AAP1345.

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