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Intersections of random sets

Part of: Geometric probability and stochastic geometry Global differential geometry

Published online by Cambridge University Press:  17 January 2022

Jacob Richey  and
Amites Sarkar
Jacob Richey*
Affiliation:
University of British Columbia
Amites Sarkar*
Affiliation:
Western Washington University
*
*Postal address: 1984 Mathematics Rd, Vancouver BC V6T 1Z2, Canada. Email: jfrichey001@gmail.com
**Postal address: 516 High Street, Bellingham WA 98225, USA. Email: amites.sarkar@wwu.edu
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Abstract

We consider a variant of a classical coverage process, the Boolean model in $\mathbb{R}^d$. Previous efforts have focused on convergence of the unoccupied region containing the origin to a well-studied limit C. We study the intersection of sets centered at points of a Poisson point process confined to the unit ball. Using a coupling between the intersection model and the original Boolean model, we show that the scaled intersection converges weakly to the same limit C. Along the way, we present some tools for studying statistics of a class of intersection models.

Keywords

Poisson processrandom tessellationCrofton cell

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 53C65: Integral geometry

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 131 - 151
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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