Asymptotic behavior of the convex hull of a stationary Gaussian process∗

Abstract

Let X = {X(t), t ∈ T} be a stationary centered Gaussian process with values in ℝ d , where the parameter set T equals ℕ or ℝ+. Let Σ t = Cov(X 0 ,X t ) be the covariance function of X, and (Ω,ℱ, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W t = conv{X u , u ∈ T ∩ [0, t]} as t → +∞ and show that under the condition Σt → 0, t→∞, the rescaled convex hull (2 ln t) −1/2 W t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ℰ associated to the covariance matrix Σ 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].