The $\beta$-Delaunay tessellation IV: Mixing properties and central limit theorems

TL;DR

This paper investigates the mixing properties of various Delaunay tessellations in Euclidean space, establishing absolute regularity and deriving central limit theorems for geometric features, with bounds on dependence decay rates.

Contribution

It introduces new concentration bounds and proves central limit theorems for geometric parameters of beta- and Gaussian Delaunay tessellations, advancing understanding of their probabilistic structure.

Findings

Beta- and Gaussian Delaunay tessellations are absolutely regular with exponential decay bounds.

Beta'-Delaunay tessellations exhibit polynomial decay in their mixing coefficients.

Central limit theorems are established for face counts and volumes of the tessellations.

Abstract

Various mixing properties of $\beta$-, $\beta'$- and Gaussian Delaunay tessellations in $\mathbb{R}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $\beta$-mixing. In the $\beta$- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the $\beta'$-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of $\beta$- and Gaussian Delaunay tessellations are established. This includes the number of $k$-dimensional faces and the $k$-volume of the $k$-sk