Limit distribution of the sample volume fraction of Boolean set

Abstract

We study the limit distribution of the volume fraction estimator ${\hat{p}}_{\lambda ,A}$ (= the Lebesgue measure of the intersection $\mathcal{X}\cap (\lambda A)$ of a random set $\mathcal{X}$ with a large observation set $\lambda A$, divided by the Lebesgue measure of $\lambda A$), as $\lambda \to \infty$, for a Boolean set $\mathcal{X}$ formed by uniformly scattered random grains $\Xi \subset {\mathbb{R}}^{\nu }$. We obtain general conditions on generic grain set $\Xi$ under which ${\hat{p}}_{\lambda ,A}$ has an $\alpha$-stable limit distribution with index $1<\alpha \le 2$. A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set observed on a large subset of a ${\nu }_0$-dimensional $(1\le {\nu }_0\le \nu -1$) hyperplane of ${\mathbb{R}}^{\nu }$.

20 pages, 2 figures