A counter-example linked to Gaussian convex hulls

TL;DR

This paper demonstrates that without weak convergence assumptions, the limit of normalized convex hulls of Gaussian elements can be any convex compact set, challenging previous convergence results.

Contribution

It shows that relaxing weak convergence assumptions allows the limit set of convex hulls to be any convex compact set, providing a counter-example to prior convergence claims.

Findings

Normalized convex hulls can converge to any convex compact set without weak convergence.

Previous convergence to the concentration ellipsoid requires weak convergence.

Counter-example invalidates the necessity of weak convergence for certain limits.

Abstract

We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov- Paulauskas (2024), its normalized convex hulls converge, with probability 1, to the concentration ellipsoid of the limiting distribution. The goal of the present note is to show that if the assumption of weak convergence of the initial sequence is relaxed, than the limit set can be an arbitrary convex compact set.