On the asymptotic form of convex hulls of Gaussian random fields

Abstract

We consider a centered Gaussian random field X = {Xt : t ∈ T} with values in a Banach space $$\mathbb{B}$$ defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of Xt is independent of t. We consider the asymptotic behavior of closed convex hulls Wn = conv{Xt : t ∈ Tn}, where (Tn) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (bn)n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(Wn), where f is some function, is also studied.