Concentration and moderate deviations for Poisson polytopes and polyhedra

TL;DR

This paper investigates the probabilistic behavior of geometric features of Poisson polytopes and polyhedra, providing sharp bounds, deviation estimates, and moderate deviation principles for various key characteristics.

Contribution

It introduces new bounds and deviation principles for geometric functionals of Poisson polytopes, including applications to Poisson hyperplane mosaics and Voronoi cells.

Findings

Sharp bounds on cumulants for geometric functionals

Exponential estimates for large deviation probabilities

Moderate deviation principles for spatial empirical measures

Abstract

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.