Advances in Applied Probability

Hard-core thinnings of germ‒grain models with power-law grain sizes

Mikko Kuronen and Lasse Leskelä

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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.

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Adv. in Appl. Probab., Volume 45, Number 3 (2013), 595-625.

First available in Project Euclid: 30 August 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes

Regular variation Boolean model marked Poisson process germ‒grain model hard-core model hard-sphere model


Kuronen, Mikko; Leskelä, Lasse. Hard-core thinnings of germ‒grain models with power-law grain sizes. Adv. in Appl. Probab. 45 (2013), no. 3, 595--625. doi:10.1239/aap/1377868531.

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