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


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


Typical weighted random simplices Z μ , μ(-2,), in a Poisson–Delaunay tessellation in 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 also mod-ϕ convergence and the large deviations behaviour of the logarithmic volume of Z μ are investigated.


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