Refined convergence for the Boolean model

TL;DR

This paper refines the understanding of convergence in the Boolean model by analyzing second-order terms and establishing a precise coupling with the Poisson line process in two dimensions.

Contribution

It introduces a detailed coupling between the Boolean model and the Poisson line process, analyzing second-order convergence terms in two dimensions.

Findings

Established a precise coupling between Boolean model and Poisson line process.

Derived directional convergence in distribution for the difference of the two sets.

Analyzed second-order terms in the convergence of the Boolean model.

Abstract

In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process is first established, a result of directional convergence in distribution for the difference of the two sets involved is derived as well.