Article

Methodology and Computing in Applied Probability

, Volume 9, Issue 4, pp 541-556

First online:

Empirical Polygon Simulation and Central Limit Theorems for the Homogenous Poisson Line Process

  • Julien MichelAffiliated withUnité de Mathématiques Pures et Appliquées Email author 
  • , Katy ParouxAffiliated withLaboratoire de Mathématiques de Besançon, UMR 6623-Université de Franche-Comté

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Abstract

For the Poisson line process the empirical polygon is defined thanks to ergodicity and laws of large numbers for some characteristics, such as the number of edges, the perimeter, the area, etc... One also has, still thanks to the ergodicity of the Poisson line process, a canonical relation between this empirical polygon and the polygon containing a given point. The aim of this paper is to provide numerical simulations for both methods: in a previous paper (Paroux, Advances in Applied Probability, 30:640–656, 1998) one of the authors proved central limit theorems for some geometrical quantities associated with this empirical Poisson polygon, we provide numerical simulations for this phenomenon which generates billions of polygons at a small computational cost. We also give another direct simulation of the polygon containing the origin, which enables us to give further values for empirical moments of some geometrical quantities than the ones that were known or computed in the litterature.

Keywords

Poisson line process central limit theorem simulation

AMS 2000 Subject Classifications

Primary 60D05 Secondary 60F05 60G55 62G30 65C50