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Hard-Core Thinnings of Germ‒Grain Models with Power-Law Grain Sizes

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

Mikko Kuronen  and
Lasse Leskelä
Mikko Kuronen*
Affiliation:
University of Jyväskylä
Lasse Leskelä*
Affiliation:
University of Jyväskylä
*
∗ Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, Jyväskylä 40014, Finland.
∗ Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, Jyväskylä 40014, Finland.
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Abstract

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Random sets with long-range dependence can be generated using a Boolean model with power-law grain sizes. We study thinnings of such Boolean models which have the hard-core property that no grains overlap in the resulting germ‒grain model. A fundamental question is whether long-range dependence is preserved under such thinnings. To answer this question, we study four natural thinnings of a Poisson germ‒grain model where the grains are spheres with a regularly varying size distribution. We show that a thinning which favors large grains preserves the slow correlation decay of the original model, whereas a thinning which favors small grains does not. Our most interesting finding concerns the case where only disjoint grains are retained, which corresponds to the well-known Matérn type-I thinning. In the resulting germ‒grain model, typical grains have exponentially small sizes, but rather surprisingly, the long-range dependence property is still present. As a byproduct, we obtain new mechanisms for generating homogeneous and isotropic random point configurations having a power-law correlation decay.

Keywords

Regular variationBoolean modelmarked Poisson processgerm‒grain modelhard-core modelhard-sphere model

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 45 , Issue 3 , September 2013 , pp. 595 - 625
Copyright
© Applied Probability Trust 

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