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

Author

Listed:
  • Christian Yann Robert

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Nicolas Chenavier

    (LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville - ULCO - Université du Littoral Côte d'Opale)

Abstract

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.

Suggested Citation

  • Christian Yann Robert & Nicolas Chenavier, 2018. "Cluster size distributions of extreme values for the Poisson–Voronoi tessellation," Post-Print hal-02006796, HAL.
    • Handle: RePEc:hal:journl:hal-02006796
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    Cited by:

    1. Pianoforte, Federico & Schulte, Matthias, 2022. "Criteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellations," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 388-422.

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