Search

Various mixing properties of β -, β ′ - and Gaussian-Delaunay tessellations in ℝ d − 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 ′ -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 }. [ABSTRACT FROM AUTHOR]

The β-Delaunay tessellation in ℝd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a βDelaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex inℝd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. [ABSTRACT FROM AUTHOR]

In this paper, we prove three fundamental types of limit theorems for the q -norm of random vectors chosen at random in an ℓ p n -ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of ℓ p n -balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Z. Kabluchko, J. Prochno and C. Thäle, High-dimensional limit theorems for random vectors in ℓ p n -balls, Commun. Contemp. Math.21(1) (2019) 1750092]. [ABSTRACT FROM AUTHOR]