The volume of simplices in high-dimensional Poisson–Delaunay tessellations
Annales Henri Lebesgue, Volume 4 (2021), pp. 121-153.

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

### Abstract

Typical weighted random simplices ${Z}_{\mu }$, $\mu \in \left(-2,\infty \right)$, in a Poisson–Delaunay tessellation in ${ℝ}^{n}$ are considered, where the weight is given by the $\left(\mu +1\right)$st power of the volume. As special cases this includes the typical ($\mu =-1$) and the usual volume-weighted ($\mu =0$) Poisson–Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of ${Z}_{\mu }$ 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 $\mu =\mu \left(n\right)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $\mu$ also mod-$\varphi$ convergence and the large deviations behaviour of the logarithmic volume of ${Z}_{\mu }$ are investigated.

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