On the asymptotic of convex hulls of Gaussian fields

TL;DR

This paper investigates the asymptotic shape of convex hulls formed by Gaussian fields in Banach spaces, showing convergence to a concentration ellipsoid under certain dependence conditions.

Contribution

It establishes the almost sure convergence of scaled convex hulls of Gaussian fields to a limit shape, extending understanding of their geometric asymptotics.

Findings

Convex hulls converge to a concentration ellipsoid in Hausdorff distance.

Results hold under weak dependence conditions.

Asymptotic behavior of expected convex hulls is characterized.

Abstract

We consider a Gaussian field $X = \{X_t, t \in T\}$ 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 $$ W_n = \conv \{X_t, t \in T_n\} $$ 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 $$ \lim_{n\rightarrow \infty} \frac{1}{b_n}\,W_n = {\cal E} $$ (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.