Gaussian Zonoids, Gaussian determinants and Gaussian random fields

TL;DR

This paper introduces Gaussian zonoids, a generalization of ellipsoids associated with Gaussian vectors, and uses them to estimate determinants and analyze zero sets of Gaussian random fields.

Contribution

It defines Gaussian zonoids for non-centered Gaussian vectors, provides explicit ellipsoid approximations, and applies these to estimate determinants and study Gaussian random fields.

Findings

Gaussian zonoids can be approximated by explicit ellipsoids.

New bounds for the expectation of Gaussian matrix determinants.

Asymptotic analysis of zero sets of Gaussian random fields.

Abstract

We study the Vitale zonoid (a convex body associated to a probability distribution) associated to a non--centered Gaussian vector. This defines a family of convex bodies, that contains and generalizes ellipsoids, which we call Gaussian zonoids. We show that each Gaussian zonoid can be approximated by an ellipsoid that we compute explicitely. We use this result to give new estimates for the expectation of the absolute value of the determinant of a non--centered Gaussian matrix in terms of mixed volume of ellipsoids. Finally, exploiting a recent link between random fields and zonoids uncovered by Stecconi and the author, we apply our results to the study of the zero set of non--centered Gaussian random fields. We show how these can be approximated by a suitable centered Gaussian random field and give a quantitative asymptotic in the limit where the variance goes to zero.