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On Laslett's transform for the Boolean model

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
V. Schmidt
A. D. Barbour*
Affiliation:
Universität Zürich
V. Schmidt*
Affiliation:
Universität Ulm
*
∗ Postal address: Department of Applied Mathematics, University of Zurich, CH-8057 Zurich, Switzerland.
∗∗ Postal address: Institute of Stochastics, Faculty of Mathematics and Economics, University of Ulm, D-89069 Ulm, Germany. Email address: schmidt@mathematik.uni-ulm.de
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Abstract

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

Keywords

stochastic geometryrandom closed setsBoolean modeltangent pointsPoisson processmartingale approach

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60G57: Random measures 60G44: Martingales with continuous parameter

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 1 - 5
Copyright
Copyright © Applied Probability Trust 2001 

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References

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