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Refined convergence for the Boolean model

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

Published online by Cambridge University Press:  01 July 2016

Pierre Calka ,
Julien Michel  and
Katy Paroux
Pierre Calka*
Affiliation:
Université Paris Descartes
Julien Michel*
Affiliation:
ENS Lyon
Katy Paroux*
Affiliation:
Université de Franche Comté and INRIA Rennes - Bretagne Atlantique
*
∗ Postal address: MAP5, UFR de Mathématiques et Informatique, Université Paris Descartes, 45 rue des Saints-Péres, 75270 Paris Cedex 06, France. Email address: pierre.calka@math-info.univ-paris5.fr
∗∗ Postal address: Unité de Mathématiques Pures et Appliquées, UMR 5669, ENS Lyon, 46 allée d'Italie, F-69364 Lyon Cedex 07, France.
∗∗∗ Postal address: Laboratoire de Mathématiques de Besançon, UMR 6623, F-25030 Besançon Cedex, France.
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Abstract

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In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.

Keywords

Poisson point processCrofton cellconvergencestochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60F99: None of the above, but in this section
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 940 - 957
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Research partially supported by the French ANR project ‘mipomodim’, grant number ANR-05-BLAN-0017.

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