Let $X_i = { X_i(t), t \in T}$ be i.i.d. copies of a centered Gaussian process $X = { X(t), t \in T}$ with values in $\mathbb{ R}^d$ defined on a separable metric space $T.$ It is supposed that $X$ is bounded. We consider the asymptotic behaviour of convex hulls $$ W_n = \conv\ { X_1(t), X_n(t), t \in T}$$ and show that with probability 1 $$ \lim_{ n\to \infty} \frac{ 1}{ \sqrt{ 2\ln n}} W_n = W $$ (in the sense of Hausdorff distance), where the limit shape $W$ is defined by the covariance structure of $X$: $W = \conv { }\{ K_t, t\in T}, K_t$ being the concentration ellipsoid of $X(t).$ The asymptotic behavior of the mathematical expectations $Ef(W_n)$, where $f$ is an homogeneous functional is also studied.
Categories: math.PR