Hard-core thinnings of germ-grain models with power-law grain sizes (Q2856027)

Consider a random closed set which can be expressed as a union of compact sets in the \(d\)-dimensional Euclidean space. The compact building blocks of the random set are called grains. This kind of random object is called the germ-grain model. The paper considers the germ-grain models with the hard-core property, that is, the grains are disjoint with probability 1. A key statistical feature of the random set is its covariance function, which describes how much more or less likely it is to find matter at a given distance from a location containing matter, compared to finding matter in an arbitrary location.NEWLINENEWLINEWhile most germ-grain models studied in the literature have a rapidly decaying covariance function, certain experimental studies in astronomy [\textit{B. J. T. Jones, V. J. Martínez, E. Saar} and \textit{V. Trimble}, ``Scaling laws in the distribution of galaxies'', Rev. Modern Phys., Vol. 76, 1211--1266 (2005)] and materials science [\textit{F. Schüth, K. S. W. Sing} and \textit{J. Weitkamp}, (eds), Handbook of porous solids, Wiley- VCH (2002)] display real-world data where the statistically estimated covariance function appears to decay exceptionally slowly, following a power law \(r^{-\beta}\) with some exponent \(\beta>0\) for large distances. When \(\beta< d\), such models are long-range dependent in the sense that \(\limsup_{r\to\infty}{{\operatorname{var}(|X\cap B_r|)}\over{r^d}} = \infty\), where \(|X\cap B_r|\) denotes the volume of the region covered by the random set \(X\) within the closed ball \(B_r\) with radius \(r\) centered at the origin, see [\textit{D. J. Daley} and \textit{D. Vere-Jones}, An introduction to the theory of point processes. Vol. II: General theory and structure. New York, NY: Springer (2008; Zbl 1159.60003)].NEWLINENEWLINEThe goal of the paper is to construct parsimonious germ-grain models having the hard-core and long-range dependence property. In the presence of long-range dependence, the requirement of parsimony, that is having a small number of model parameters, is especially important because the long-range dependence tends to reduce the robustness of the statistical estimators of model parameters [\textit{A. Clauset, C. R. Shalizi} and \textit{M. E. J. Newman}, SIAM Rev. 51, No. 4, 661--703 (2009; Zbl 1176.62001)]. This paper may be seen as a continuation of the works by \textit{M. Månsson} and \textit{M. Rudemo} [Adv. Appl. Probab. 34, No. 4, 718--738 (2002; Zbl 1020.60004)] and \textit{J. Andersson, O. Häggström} and \textit{M. Månsson} [Electron. Commun. Probab. 11, 78--88 (2006; Zbl 1111.60007)].NEWLINENEWLINEThe rest of the paper is organized as follows. Section 2 summarizes preliminaries on random Boolean models. In Section 3, the authors introduce a weight-based thinning mechanism which produces hard-core germ-grain models from Boolean models and list formulae for the second-order characteristics of the models so obtained. Section 4 contains a long-range analysis of the second-order characteristics of the previous section. The main results of the paper are proved in Sections 5--8. Section 9 concludes the paper.