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August, 1994 Uniqueness of Unbounded Occupied and Vacant Components in Boolean Models
Ronald Meester, Rahul Roy
Ann. Appl. Probab. 4(3): 933-951 (August, 1994). DOI: 10.1214/aoap/1177004978


We consider Boolean models in $d$-dimensional Euclidean space. Each point of a stationary, ergodic point process is the center of a ball with random radius. In this way, the space is partitioned into an occupied and a vacant region. We are interested in the number of unbounded occupied or vacant components that can coexist. We show that under very general conditions on the distribution of the radius random variable, there can be at most one unbounded component of each type. In case the point process is Poisson, we obtain uniqueness of the unbounded components without imposing any condition at all. Although we do not prove the necessity of the conditions to prove uniqueness, we obtain examples of stationary, ergodic point processes where the unbounded components are not unique when the conditions are violated. Finally, we discuss more general random shapes than just balls which are centered at the points of the point process.


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Ronald Meester. Rahul Roy. "Uniqueness of Unbounded Occupied and Vacant Components in Boolean Models." Ann. Appl. Probab. 4 (3) 933 - 951, August, 1994.


Published: August, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0812.60093
MathSciNet: MR1284992
Digital Object Identifier: 10.1214/aoap/1177004978

Primary: 60K35

Keywords: Boolean models , continuum percolation , Point processes , uniqueness

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 3 • August, 1994
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