On convex hull of Gaussian samples

2010/04/27 by Yu. I. Davydov, Yu. Davydov, Davydov, Yu.
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1004.4908

10 pages

arxiv created 2010/04/27 · openalex publication_date 2010/04/27 · arxiv updated 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X_i = X_i(t), t ∈ T be i.i.d. copies of a centered Gaussian process X = X(t), t ∈ T with values in ℝ^d defined on a separable metric space T. It is supposed that X is bounded. We consider the asymptotic behaviour of convex hulls W_n = \conv X₁(t), X_n(t), t ∈ T and show that with probability 1 lim_{n→ ∞} (1)/(√(2ln n)) W_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∈ T, K_t being the concentration ellipsoid of X(t). The asymptotic behavior of the mathematical expectations Ef(W_n), where f is an homogeneous functional is also studied.

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