Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T07:02:41.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Refined convergence for the Boolean model

Part of: Stochastic processes Geometric probability and stochastic geometry 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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Calka, P. (2002). The distributions of the smallest disks containing the Poisson–Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Prob. 34, 702–717.Google Scholar
Calka, P. (2003). Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson–Voronoi tessellation and a Poisson line process. Adv. Appl. Prob. 35, 551–562.Google Scholar
Hall, P. (1985). Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrscheinlichkeitsth. 70, 237–261.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Heinrich, L., Schmidt, H. and Schmidt, V. (2006). Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Prob. 16, 919–950.Google Scholar
Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Prob. 32, 1140–1167.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Michel, J. and Paroux, K. (2003). Local convergence of the Boolean shell model towards the thick Poisson hyperplane process in the Euclidean space. Adv. Appl. Prob. 35, 354–361.Google Scholar
Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 901–907.Google Scholar
Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 1157–1160.CrossRefGoogle ScholarPubMed
Molchanov, I. S. (1996). A limit theorem for scaled vacancies of the Boolean model. Stoch. Stoch. Reports 58, 45–65.Google Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
Paroux, K. (1997). Théorèmes centraux limites pour les processus poissoniens de droites dans le plan et questions de convergence pour le modèle booléen de l'espace euclidien. . Université Lyon 1.Google Scholar
Paroux, K. (1998). Quelques théorèmes centraux limites pour les processus Poissoniens de droites dans le plan. Adv. Appl. Prob. 30, 640–656.Google Scholar
Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar