A counter-example linked to Gaussian convex hulls

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.