Intersections of randomly translated sets

TL;DR

This paper studies the asymptotic behavior of intersections of randomly translated sets in Euclidean space, showing convergence to a Poisson hyperplane tessellation determined by geometric and measure-theoretic properties.

Contribution

It introduces a novel limit theorem for the scaled complement of intersections of random translations of a convex set, linking it to Poisson hyperplane tessellations.

Findings

Convergence of scaled complements to a Poisson hyperplane tessellation.

Dependence on curvature measure and boundary density of the original set.

Provides a probabilistic geometric limit theorem.

Abstract

Let $\Xi_n=\{\xi_1,\dots,\xi_n\}$ be a sample of $n$ independent points distributed in a regular closed element $K$ of the extended convex ring in $\mathbb{R}^d$ according to a probability measure $\mu$ on $K$, admitting a density function. We consider random sets generated from the intersection of the translations of $K$ by elements of $\Xi_n$, as $X_n=\bigcap_{i=1}^n (K-\xi_i)$. This work aims to show that the scaled closure of the complement of $X_n$ as $n\to\infty$ converges in distribution to the closure of the complement zero cell of a Poisson hyperplane tessellation whose distribution is determined by the curvature measure of $K$ and the behaviour of the density of $\mu$ near the boundary of $K$.