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Bounds for clump size characteristics in the Boolean model

Part of: Geometric probability and stochastic geometry Markov processes

Published online by Cambridge University Press:  01 July 2016

Bartłomiej Błaszczyszyn ,
Christian Rau  and
Volker Schmidt
Bartłomiej Błaszczyszyn*
Affiliation:
University of Wrocław
Christian Rau*
Affiliation:
University of Ulm
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: blaszcz@math.uni.wroc.pl
∗∗ Postal address: Institute of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
∗∗ Postal address: Institute of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
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Abstract

A random spatial coverage process whose generating point process is homogeneous Poisson, and whose attached random sets are independent and identically distributed, is called a Boolean model. Motivated by Błaszczyszyn et al. [1], distributional and higher moment properties of the size of clumps (connected clusters of overlapping sets) in this model are derived. This provides some complements to the result on the finiteness of the first moment presented in Hall [3]. The key idea is to construct a certain coupling process for a multitype branching process that dominates the clump size.

Keywords

Boolean modelclumpshigher momentsexponential momentmultitype branching processcoupling

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J85: Applications of branching processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 910 - 928
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Supported by KBN under grant 2 P03A 046 08.

References

Błaszczyszyn, B., Merzbach, E. and Schmidt, V. (1997). A note on expansion for functionals of spatial marked point processes. Statist. Prob. Lett. 36, 299–306.Google Scholar
Good, I. J. (1960). Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes. Proc. Camb. Phil. Soc. 56, 367–380.Google Scholar
Hall, P. (1985). On continuum percolation. Ann. Prob. 13, 1250–1266.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
Otter, R. (1949). The multiplicative process. Ann. Math. Statist. 20, 206–224.Google Scholar
Rau, C. (1997). Series expansion for characteristics of Boolean models. , University of Ulm, Germany.Google Scholar
Satake, I. (1975). Linear Algebra (Pure and Applied Mathematics 29). Marcel Dekker, New York.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester.Google Scholar