Coverage of space in Boolean models

Abstract

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

Published at http://dx.doi.org/10.1214/074921706000000158 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)