Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-11T05:05:57.480Z Has data issue: false hasContentIssue false
  • English
  • Français

Local convergence of the Boolean shell model towards the thick Poisson hyperplane process in the Euclidean space

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Julien Michel  and
Katy Paroux
Julien Michel*
Affiliation:
ENS Lyon
Katy Paroux*
Affiliation:
Université de Franche Comté
*
∗ Postal address: Unité de Mathématiques Pures et Appliquées, UMR 5669, F-69364 Lyon Cedex 07, France. Email address: [email protected]
∗∗ Postal address: Laboratoire de Mathématiques de Besançon, UMR 6623, F-25030 Besançon Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we prove local convergence for a Boolean model of shells conditioned by the noncovering of the origin towards the thick hyperplane Poisson process in the Euclidean space. The existing results of Hall as well as the convergence theorems proved by Paroux or Molchanov concerned the zero-width process and the connected component of the unfilled region of the origin. Our results deal with the convergence in any given window of the space, with the earlier results of Paroux and Molchanov as a corollary.

Keywords

Stochastic geometrypoint process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 354 - 361
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Calka, P. (2002). A remark about a result of R. E. Miles concerning the Poissonian tessellation of the Euclidean space. Preprint 02-01, LaPCS, Université Lyon 1.Google Scholar
[2] Hall, P. (1985). Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrscheinlichkeitsth 70, 237–261.Google Scholar
[3] Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
[4] Miles, R. (1964). Random polygons determined by random lines in a plane I. Proc. Nat. Acad. Sci. USA 52, 901–907.CrossRefGoogle Scholar
[5] Miles, R. (1964). Random polygons determined by random lines in a plane II. Proc. Nat. Acad. Sci. USA 52, 1157–1160.CrossRefGoogle Scholar
[6] Molchanov, I. (1996). A limit theorem for scaled vacancies of the Boolean model. Stoch. Stoch. Reports 58, 45–65.CrossRefGoogle Scholar
[7] 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. Doctoral Thesis, Université Lyon 1.Google Scholar
[8] Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar