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

Abstract

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.

Contact distribution in a thinned Boolean model with power-law radii

“…It is interesting that, as it was shown in Kuronen and Leskelä (2013), when the radii of the balls are regularly varying as in (1.7), the covariance function of the thinned grain cover (1.6) has a power-like decay under all of the above thinning mechanisms apart from the case of heavy small balls.…”

“…We refer the reader to Resnick (2007) for information on regular varying tails. Under the assumption (1.7) of regular variation, Kuronen and Leskelä (2013) discovered appearance of power-like decay of the covariance function of the thinned grain cover (1.6), under three out of four choices of the joint law G in (1.1) they considered; we will return to these choices in a moment. In this paper we are interested in the contact distribution for the thinned Boolean model described above.…”

“…Informally, for every pair of different balls, B Rn (X n ) and B Rm (X m ) with a non-empty intersection, the ball B Rn (X n ) gets deleted if W n ≤ W m ; this procedure deletes both balls if W n = W m . To be a bit more formal, we use the notation borrowed from Kuronen and Leskelä (2013): let (1.4) N x,r,w = (x , r , w ) ∈ R d × R + × R + \{x, r, w} : B r (x ) ∩ B r (x) = ∅ ;…”

“…This model was studied in Mansson and Rudemo (2002) and Anderson et al (2006). Our inspiration for the present work came from the paper of Kuronen and Leskelä (2013), and we refer the reader to this paper for an illuminating discussion of the importance and applications of hard-core germ-grain models. Specifically, the latter paper considers the case of power law grain sizes; in our notation we can describe this setup as follows.…”

“…Specifically, the latter paper considers the case of power law grain sizes; in our notation we can describe this setup as follows. Recalling that we denote by F the marginal distribution of the probability measure G in (1.1) corresponding to the random radius of a Poisson ball, the power law distribution of the grain sizes in Kuronen and Leskelä (2013) is the assumption of regular variation of the tail…”