More on the convergence of Gaussian convex hulls

Abstract: A "law of large numbers" for consecutive convex hulls for weakly dependent Gaussian sequences ${X_n}$, having the same marginal distribution, is extended to the case when the sequence ${X_n}$ has a weak limit. Let $\mathbb{B}$ be a separable Banach space with a conjugate space $\mathbb{B}^\ast$. Let ${X_n}$ be a centered $\mathbb{B}$-valued Gaussian sequence satisfying two conditions: 1) $X_n \Rightarrow X\;\;$ and 2) For every $x^* \in \mathbb{B}^\ast$ $$ \lim_ {n,m, |n-m|\rightarrow \infty}E\langle X_n, x^*\rangle \langle X_m, x^*\rangle\;\; = \;\;0. $$ Then with probability 1 the normalized convex hulls $$ W_n = \frac{1}{(2\ln n)^{1/2}}\,{\rm conv} {\,X_1,\ldots,X_{n}\,} $$ converge in Hausdorff distance to the concentration ellipsoid of a limit Gaussian $\mathbb{B}$-valued random element $X.$ In addition, some related questions are discussed.