ON CONVEX HULL OF GAUSSIAN SAMPLES

Let X-i = {X-i(t), t is an element of T} be i. i. d. copies of a centered Gaussian process X = {X(t), t is an element of T} with values in R-d defined on a separable metric space T. It is supposed that X is bounded. We consider the asymptotic behavior of convex hulls W-n = conv{X-1(t),..., X-n(t), t is an element of T} and show that, with probability 1, lim n ->infinity 1/root 2ln nW(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 is an element of T}, K-t being the concentration ellipsoid of X(t). We also study the asymptotic behavior of the mathematical expectations Ef(W-n), where f is an homogeneous functional.