Institute of Mathematical Statistics Lecture Notes - Monograph Series

Coverage of space in Boolean models

Rahul Roy

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For a marked point process $\{(x_i, S_i)_{i \geq 1}\}$ with $\{x_i \in \Lambda: i \geq 1\}$ being a point process on $\Lambda \subseteq \mathbb R^d$ and $\{S_i \subseteq R^d: i \geq 1\}$ being random sets consider the region $C= \cup_{i \geq 1} (x_i + S_i)$. This is the {\it covered}\/ region obtained from the Boolean model$\{(x_i + S_i): i \geq 1\}$. The Boolean model is said to be {\it completely covered}\/ if $\Lambda \subseteq C$ almost surely. If $\Lambda$ is an infinite set such that ${\bf s} + \Lambda \subseteq \Lambda$ for all ${\bf s} \in \Lambda$ (e.g. the orthant), then the Boolean model is said to be {\it eventually covered}\/ if ${\bf t} + \Lambda \subseteq C$ for some ${\bf t}$ almost surely. We discuss the issues of coverage when $\Lambda$ is $\mathbb R^d$ and when $\Lambda$ is $[0,\infty)^d$.

Chapter information

Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, eds., Dynamics & Stochastics: Festschrift in honor of M. S. Keane (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 119-127

First available in Project Euclid: 28 November 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 05C40: Connectivity
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Poisson process Boolean model coverage

Copyright © 2006, Institute of Mathematical Statistics


Roy, Rahul. Coverage of space in Boolean models. Dynamics & Stochastics, 119--127, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000158.

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