Intersections of random sets

TL;DR

This paper investigates the behavior of intersections of random sets in a boolean model, demonstrating their convergence to a known limit and providing tools for analyzing such intersection models.

Contribution

It introduces a coupling method to show the weak convergence of scaled intersections to the same limit as the boolean model, extending understanding of intersection statistics.

Findings

Scaled intersections converge weakly to the limit C

Coupling technique links intersection models to boolean models

Tools developed for analyzing intersection statistics

Abstract

We consider a variant of a classical coverage process, the boolean model in $\mathbb{R}^d$. Previous efforts have focused on convergence of the unoccupied region containing the origin to a well studied limit $C$. We study the intersection of sets centered at points of a Poisson point process confined to the unit ball. Using a coupling between the intersection model and the original boolean model, we show that the scaled intersection converges weakly to the same limit $C$. Along the way, we present some tools for studying statistics of a class of intersection models.