Asymptotic behavior of the convex hull of a stationary Gaussian process

Let X = {X(t), t a T} be a stationary centered Gaussian process with values in a"e (d) , where the parameter set T equals a"center dot or a"e+. Let I pound (t) = Cov(X (0) ,X (t) ) be the covariance function of X, and (Omega,F, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W (t) = conv{X (u) , u a T a (c) [0, t]} as t -> +a and show that under the condition I t pound -> 0, t -> a, the rescaled convex hull (2 ln t) (-1/2) W (t) converges almost surely (in the sense of Hausdorff distance) to an ellipsoid a"degrees associated to the covariance matrix I pound (0). The asymptotic behavior of the mathematical expectations E f(W (t) ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171-179, 2011].