Rescaling the Vacancy of a Boolean Coverage Process

Abstract

In his recent book [H], Peter Hall gives an encyclopaedic account of the theory of the class of random sets known as Boolean coverage processes. We will define this class rigorously in §2, but for the moment we give an intuitive description. Let Π be a homogeneous Poisson point process on ℝ^d which we enumerate as $\Pi =\left \{ \xi _\textup{i} \right \}^\infty _{ \textup{i = }1 }$. $\left \{ \textup{S}_\textup{i} \right \}^\infty _{\textup{i = }1}$be an independent sequence of independent, identically distributed, random open sets. The Boolean coverage process constructed from the collection of centres or germs, {ξ_i}, and the collection of shapes or grains, {S_i}, is the random open set $\textup{U = U}_\textup{i }(\xi_\textup{i}+\textup{S}_\textup{i})$.

Research carried out at the University of Virginia and supported in part by NSF Grant DMS-8701212.