Gaussian approximation for Extreme Points in Laguerre tessellations

TL;DR

This paper establishes quantitative Gaussian approximation results for the number of extreme points in Poisson-Laguerre tessellations, extending previous work and providing new bounds for multiple models in growing observation windows.

Contribution

It introduces new quantitative central limit theorems with optimal convergence rates for extreme points in three Poisson-Laguerre tessellation models, including the first such results for the $eta'$ and Gaussian models.

Findings

Derived bounds improve previous results for the $eta$-model.

First quantitative CLTs for the $eta'$- and Gaussian-Voronoi tessellations.

Optimal rates of convergence in growing windows.

Abstract

We consider Gaussian approximation in three particular models of Poisson-Laguerre tessellations, namely, the $\beta$-, $\beta'$- and Gaussian-Voronoi tessellations. The tessellations are constructed based on inhomogeneous Poisson point processes in space-time $\mathbb{R}^d \times \mathbb{R}$, where some of the points of the process give rise to a cell in $\mathbb{R}^d$, known as extreme points, while the other points produce an empty cell. Using the notion of region-stabilization, we derive quantitative central limit theorems with presumably optimal rates of convergence for the number of extreme points of $\beta$-, $\beta'$- and Gaussian-Voronoi tessellations in a growing window $W_n=[-n,n]^d$ as $n\to\infty$. Our bounds improve and extend previously known results by Schreiber and Yukich (2008) for the $\beta$-model, and are the first quantitative results for the $\beta'$- and Gaussian models.