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

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  15 September 2017

Yinghua Dong  and
Gennady Samorodnitsky
Yinghua Dong*
Affiliation:
Nanjing University of Information Science and Technology
Gennady Samorodnitsky*
Affiliation:
Cornell University
*
* Postal address: College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China. Email address: dongyinghua1@163.com
** Postal address: School of Operations Research and Information Engineering and Department of Statistical Science, Cornell University, Ithaca, NY 14853, USA. Email address: gs18@cornell.edu
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Abstract

We consider a weighted stationary spherical Boolean model in ℝd to which a Matérn-type thinning is applied. Assuming that the radii of the balls in the Boolean model have regularly varying tails, we establish the asymptotic behavior of the tail of the contact distribution of the thinned germ–grain model under four different thinning procedures of the original model.

Keywords

Boolean modelhard-core thinningpower tailcontact distributionempty space

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G57: Random measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 670 - 684
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] 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
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. CrossRefGoogle Scholar
[3] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd edn. John Wiley, Chichester. Google Scholar
[4] Hug, D., Last, G. and Weil, W. (2002). A survey on contact distributions. In Morphology of Condensed Matter (Lecture Notes Physics 600), Springer, Berlin, pp. 317–357. Google Scholar
[5] Kuronen, M. and Leskelä, L. (2013). Hard-core thinnings of germ-grain models with power-law grain sizes. Adv. Appl. Prob. 45, 595–625. Google Scholar
[6] Månsson, M. and Rudemo, M. (2002). Random patterns of nonoverlapping convex grains. Adv. Appl. Prob. 34, 718–738. Google Scholar
[7] Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitstheorie. 9, 36–58. Google Scholar
[8] Nguyen, T. V. and Baccelli, F. (2013). On the generating functionals of a class of random packing point processes. Preprint. Available at http://arxiv.org/abs/1311/4967. Google Scholar
[9] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York. Google Scholar
[10] Teichmann, J., Ballani, F. and van den Boogaart, K. G. (2013). Generalizations of Matérn's hard-core point processes. Spatial Statist. 3, 33–53. Google Scholar