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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 

References

Andersson, J., Häggström, O. and Månsson, M. (2006). The volume fraction of a non-overlapping germ-grain model. Electron. Commun. Prob. 11, 78–88.Google Scholar
Baccelli, F. and Błaszczyszyn, B. (2009). Stochastic Geometry and Wireless Networks, Vol. II. NoW Publishers. Available at http://hal.inria.fr/inria-00403040.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Böhm, S. and Schmidt, V. (2003). Palm representation and approximation of the covariance of random closed sets. Adv. Appl. Prob. 35, 295–302.Google Scholar
Clauset, A., Shalizi, C. R. and Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Rev. 51, 661–703.CrossRefGoogle Scholar
Daley, D. J. (1999). The Hurst index of long-range dependent renewal processes. Ann. Prob. 27, 2035–2041.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Daley, D. J. and Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stoch. Process. Appl. 70, 265–282.CrossRefGoogle Scholar
Demichel, Y., Estrade, A., Kratz, M. and Samorodnitsky, G. (2011). How fast can the chord length distribution decay? Adv. Appl. Prob. 43, 504–523.Google Scholar
Haenggi, M. (2011). Mean interference in hard-core wireless networks. IEEE Commun. Lett. 15, 792–794.CrossRefGoogle Scholar
Jones, B. J. T., Martı´nez, V. J., Saar, E. and Trimble, V. (2005). Scaling laws in the distribution of galaxies. Rev. Modern Phys. 76, 1211–1266.Google Scholar
Kaj, I., Leskelä, L., Norros, I. and Schmidt, V. (2007). Scaling limits for random fields with long-range dependence. Ann. Prob. 35, 528–550.Google Scholar
Kulik, R. and Szekli, R. (2001). Sufficient conditions for long-range count dependence of stationary point processes on the real line. J. Appl. Prob. 38, 570–581.Google Scholar
Månsson, M. and Rudemo, M. (2002). Random patterns of nonoverlapping convex grains. Adv. Appl. Prob. 34, 718–738.Google Scholar
Matérn, B. (1960). Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations. Meddelanden Fran Statens Skogsforskningsinstitut, Stockholm.Google Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
Nguyen, T. V. and Baccelli, F. (2012). Generating functionals of random packing point processes: from hard-core to carrier sensing. Preprint. Available at http://arxiv.org/abs/1202.0225v2.Google Scholar
Ohser, J. and Mücklich, F. (2000). Statistical Analysis of Microstructures in Materials Science. John Wiley.Google Scholar
Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Systems 1, 163–257.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Schäth, F., Sing, K. S. W. and Weitkamp, J. (eds) (2002). Handbook of Porous Solids. Wiley-VCH.Google Scholar
Snethlage, M., Martı´nez, V. J., Stoyan, D. and Saar, E. (2002). Point field models for the galaxy point pattern. Astron. Astrophys. 388, 758–765.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Vamvakos, S. and Anantharam, V. (1998). On the departure process of a leaky bucket system with long-range dependent input traffic. Queueing Systems 28, 191–214.CrossRefGoogle Scholar