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 covered region obtained from the Boolean model $\{(x_i+S_i):i\geq 1\}$. The Boolean model is said to be 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 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 $[0,\infty)^d$.