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

Gusakova, Anna; Kabluchko, Zakhar; Thäle, Christoph

Abstract

Various mixing properties of β-, β' - and Gaussian Delaunay tessellations in Rd-1 are studied. It is shown that these tessellation models are absolutely regular, or β-mixing. In the β- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the β 0 -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 β- and Gaussian Delaunay tessellations are established. This includes the number of k-dimensional faces and the k-volume of the k-skeleton for k ∈ {0, 1, . . . , d − 1} MSC: 52A22, 52B11, 53C65, 60D05, 60F05.