Extremal behavior of large cells in the Poisson hyperplane mosaic

TL;DR

This paper investigates the asymptotic distribution of large cells in a Poisson hyperplane mosaic, establishing a Poisson limit theorem for their centers based on size measures like inradius and intrinsic volumes.

Contribution

It generalizes previous results by proving a Poisson limit theorem for large cells in higher dimensions and for various size measures, extending existing approximation techniques.

Findings

Established a Poisson limit theorem for large cell centers

Extended approximation results to a broader class of size measures

Generalized previous work to higher dimensions and different intrinsic volumes

Abstract

We study the asymptotic behavior of a size-marked point process of centers of large cells in a stationary and isotropic Poisson hyperplane mosaic in dimension $d \ge 2$. The sizes of the cells are measured by their inradius or their $k$th intrinsic volume ($k \ge 2$), for example. We prove a Poisson limit theorem for this process in Kantorovich-Rubinstein distance and thereby generalize a result in Chenavier and Hemsley (2016) in various directions. Our proof is based on a general Poisson process approximation result that extends a theorem in Bobrowski, Schulte and Yogeshwaran (2021).