  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.
