The volume of simplices in high-dimensional Poisson–Delaunay tessellations

Abstract

Typical weighted random simplices Z µ , µ ∈ (−2, ∞), in a Poisson-Delaunay tessellation in R n are considered, where the weight is given by the (µ + 1)st power of the volume. As special cases this includes the typical (µ = −1) and the usual volume-weighted (µ = 0) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of Z µ satisfies a central limit theorem in high dimensions, that is, as n → ∞. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight µ = µ(n) to depend on the dimension n as well. A number of special cases are discussed separately. For fixed µ

The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions

“…We emphasize that in the special case β = −1, which corresponds to the classical Poisson-Delaunay tessellation, this covers previous results from [14]. We also remark in this context that central limit theorems for other functionals of β-Delaunay tessellations for fixed space dimensions but in increasing observation windows will be derived in part IV of this paper.…”

“…We remark that although [9] studies very general models with so-called gamma type moments, our random variables do not precisely fit into this framework. Our argument closely follows the one in [14] with suitable modifications and adaptions, of course. Before we state the main result of this section let us recall the definition of the Barnes G-function.…”

“…The motivation for such a study is driven by the fact that probabilistic limit theorems for convex bodies in high dimensions is a central theme in the branch of mathematics called Asymptotic Geometric Analysis. The behaviour of the logarithmic volume of random convex bodies and especially of random simplices in high dimensions has recently been studied in stochastic geometry in [2,11,14]. We continue this line of research by providing central limit theorems as well as moderate and large deviation principles for Y β,ν,d := log Vol(Z β,ν ), as d → ∞.…”

“…This will be the case for our application presented below. Mod-φ convergence for the log-volume of different models of random simplices was recently studied in [9,11,14]. We remark that although [9] studies very general models with so-called gamma type moments, our random variables do not precisely fit into this framework.…”

“…It should be noted, that by Remark 2. . This case was considered in [14] and the case of fixed β can be treated analogously. In this article we focus from now on on the particularly interesting situation when ν is some fixed number.…”