Gaussian polytopes: a cumulant-based approach

TL;DR

This paper introduces a cumulant-based approach to analyze Gaussian polytopes, leading to new concentration inequalities, limit theorems, and deviation principles for their geometric properties.

Contribution

It develops a novel cumulant and large deviation framework for Gaussian polytopes, providing new probabilistic bounds and limit results.

Findings

New concentration inequalities for Gaussian polytopes

Central limit theorems for volume and face numbers

Moderate deviation principles established

Abstract

The random convex hull of a Poisson point process in $\mathbb{R}^d$ whose intensity measure is a multiple of the standard Gaussian measure on $\mathbb{R}^d$ is investigated. The purpose of this paper is to invent a new viewpoint on these Gaussian polytopes that is based on cumulants and the general large deviation theory of Saulis and Statulevi\v{c}ius. This leads to new and powerful concentration inequalities, moment bounds, Marcinkiewicz-Zygmund-type strong laws of large numbers, central limit theorems and moderate deviation principles for the volume and the face numbers. Corresponding results are also derived for the empirical measures induced by these key geometric functionals, taking thereby care of their spatial profiles.