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

Abstract

Various mixing properties of $\beta$-, ${\beta }^{\prime }$- 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 ${\beta }^{\prime }$-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

References in corpus (1)

• Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points