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

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