On the asymptotic of convex hulls of Gaussian fields

2012/10/20 by Youri Davydov, Davydov, Youri, Vigantas Paulauskas +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1210.5590

arxiv created 2012/10/20 · openalex publication_date 2012/10/20 · arxiv updated 2012/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Related

• On convex hull of Gaussian samples
• More on the convergence of Gaussian convex hulls
• Variance asymptotics and scaling limits for Gaussian Polytopes
• Extremes of Gaussian random fields with non-additive dependence structure
• On the tightness of Gaussian concentration for convex functions