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

Gusakova, Anna; Thäle, Christoph

doi:10.5802/ahl.68

Gusakova, Anna; Thäle, Christoph

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

Annales Henri Lebesgue, Volume 4 (2021), pp. 121-153.

Metadata

DOI10.5802/ahl.68

Keywords Berry–Esseen bound, central limit theorem, cumulants, high dimensions, mod-$\phi $ convergence, moderate deviations, large deviations, random simplex, Poisson–Delaunay tessellation, stochastic geometry

Abstract

Typical weighted random simplices $Z_{μ}$ , $μ \in (- 2, \infty)$ , 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 $n \to \infty$ . 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 $μ$ also mod- $ϕ$ convergence and the large deviations behaviour of the logarithmic volume of $Z_{μ}$ are investigated.

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