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

TL;DR

This paper studies the distribution of large-value cells in Poisson-Voronoi tessellations, characterizing their clustering behavior and providing methods to compute extremal indices and cluster sizes through simulations.

Contribution

It introduces a new framework for analyzing the asymptotic cluster size distribution of extreme cells in Poisson-Voronoi tessellations, including efficient computation techniques.

Findings

Convergence conditions for exceedance point processes to a homogeneous compound Poisson process.

Characterization of the asymptotic cluster size distribution using Palm calculus.

Simulation-based computation of extremal index and cluster size probabilities.

Abstract

We consider the Voronoi tessellation based on a homogeneous Poisson point process in $\mathbf{R}^{d}$. 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.