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Intersections of random sets

Jacob Richey, Amites Sarkar

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.

Intersections of random sets

Abstract

We consider a variant of a classical coverage process, the boolean model in . Previous efforts have focused on convergence of the unoccupied region containing the origin to a well studied limit . 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 . Along the way, we present some tools for studying statistics of a class of intersection models.

Paper Structure

This paper contains 5 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Random variables in the space of closed sets
  3. Overview of results
  4. Statistics for the intersection model
  5. Ball and hyperplane intersection models

Key Result

Theorem 1.1

Let $C_0^\lambda$, $D_0^\lambda$ and $E_0^\lambda$ denote the Crofton cell in the boolean model (with $S=\mathbb{B}$), and the connected components containing the origin in the hyperplane and sphere tessellation models, respectively. These three models have a common scaling limit in distribution: where $S_d$ is the surface area measure of $\mathbb{S} = \partial \mathbb{B}$, $C$ is the law of the

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 3.1
  • proof