## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 48, 2006, 119-127

### Coverage of space in Boolean models

#### Abstract

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

**Source***Dynamics & Stochastics: Festschrift in honor of M. S. Keane* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196285814

**Digital Object Identifier**

doi:10.1214/074921706000000158

**Mathematical Reviews number (MathSciNet)**

MR2306194

**Zentralblatt MATH identifier**

1123.60007

**Subjects**

Primary: 05C80: Random graphs [See also 60B20] 05C40: Connectivity

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Poisson process Boolean model coverage

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Roy, Rahul. Coverage of space in Boolean models. Dynamics & Stochastics, 119--127, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000158. https://projecteuclid.org/euclid.lnms/1196285814