On convex hull of Gaussian samples

TL;DR

This paper investigates the asymptotic shape of convex hulls formed by Gaussian samples in a metric space, revealing they converge to a shape determined by the process's covariance structure.

Contribution

It establishes the almost sure convergence of scaled convex hulls of Gaussian samples to a shape defined by the covariance ellipsoids, extending understanding of Gaussian sample geometry.

Findings

Convex hulls scaled by √(2ln n) converge to a deterministic shape.

Limit shape is the convex hull of covariance ellipsoids.

Asymptotic expectations of homogeneous functionals are characterized.

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 $$ \lim_{n\to \infty} \frac{1}{\sqrt{2\ln n}} W_n = W $$ (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.