Limit distribution of the sample volume fraction of Boolean set

TL;DR

This paper investigates the asymptotic distribution of the volume fraction estimator for Boolean models with random grains, establishing conditions for stable limit distributions as observation size grows large.

Contribution

It provides general conditions under which the volume fraction estimator converges to an $ ext{alpha}$-stable distribution for Boolean models with homothetic grains.

Findings

The estimator has an $ ext{alpha}$-stable limit distribution with index $1< ext{alpha} extless 2$.

Introduces a class of Boolean models satisfying the conditions for stable limits.

Discusses limit distributions on lower-dimensional hyperplanes.

Abstract

We study the limit distribution of the volume fraction estimator $\widehat p_{\lambda, A}$ (= 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 $\widehat p_{\lambda, A}$ 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 $\mathbb{R}^\nu$.