On convex hull of Gaussian samples

Abstract

Let $X_i=X_i(t),t\in T$ be i.i.d. copies of a centered Gaussian process $X=X(t),t\in T$ with values in ${\mathbb{R}}^d$ defined on a separable metric space $T.$ It is supposed that $X$ is bounded. We consider the asymptotic behaviour of convex hulls $$ W_n = \conv\ {X_1(t), X_n(t), t \in T}$$ and show that with probability 1 $$\underset{n\to \infty }{\lim }\frac{1}{\sqrt{2\ln n}}{W}_n=W$$ (in the sense of Hausdorff distance), where the limit shape $W$ is defined by the covariance structure of $X$: $W = \conv {}\{K_t, t\in T}, K_t$ 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.

10 pages

References in corpus (2)

• Random Convex Hulls and Extreme Value Statistics
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology