On the asymptotic of convex hulls of Gaussian fields

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We consider a Gaussian field X = {Xₜ, t ∈ T} with values in a Banach space B defined on a parametric set T equal to Rᵐ or Zᵐ. It is supposed that the distribution P of Xₜ is independent of t. We consider the asymptotic behavior of closed convex hulls Wₙ = {Xₜ, t ∈ Tₙ} where (Tₙ) is an increasing sequence of subsets of T and we show that under some conditions of the weak dependence with probability 1 ₙ→ ∞ 1/bₙWₙ = E (in the sense of Hausdorff distance), where the limit shape E is the concentration ellipsoid of P. The asymptotic behavior of the mathematical expectations Ef(Wₙ), where f is an homogeneous function is also studied.