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Extremes for the inradius in the Poisson line tessellation

Part of: Nonparametric inference Stochastic processes Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  10 June 2016

Nicolas Chenavier  and
Ross Hemsley
Nicolas Chenavier*
Affiliation:
Université du Littoral Côte d'Opale
Ross Hemsley*
Affiliation:
Inria
*
* Postal address: LMPA Joseph Liouville, Université du Littoral Côte d'Opale, 50 rue Ferdinand Buisson, BP 699, F-62228 Calais Cedex, France. Email address: nicolas.chevavier@lmpa.univ-littoral.fr
** Postal address: Inria, BP 93, 06902 Sophia-Antipolis Cedex, France.
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Abstract

A Poisson line tessellation is observed in the window Wρ := B(0, π-1/2ρ1/2) for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using the Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to ∞. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

Keywords

Line tessellationPoisson point processextreme valueorder statistic

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 62G32: Statistics of extreme values; tail inference
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 544 - 573
Copyright
Copyright © Applied Probability Trust 2016 

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