Minimax of an n-dimensional Brownian motion

TL;DR

This paper demonstrates that for high-dimensional Brownian motion, the convex hull over a specific time interval almost surely excludes the origin, providing insights into the minimax of associated Gaussian processes.

Contribution

It establishes a probabilistic bound showing the convex hull of high-dimensional Brownian motion typically does not contain the origin, advancing understanding of Gaussian process minimax behavior.

Findings

Convex hull of high-dimensional Brownian motion often excludes the origin.

Provides probabilistic estimates for the minimax of Gaussian processes.

Results hold for sufficiently large dimensions n.

Abstract

For some absolute constants $c$, $n_0$ and any $n\geq n_0$, we show that with probability close to one the convex hull of the $n$-dimensional Brownian motion ${\rm conv}\{BM_n(t):\, t\in[1,2^{cn}]\}$ does not contain the origin. The result can be interpreted as an estimate of the minimax of the Gaussian process $\{ \langle \bar{u},BM_n(t)\rangle,\, \bar{u}\in S^{n-1},\, t\in [1,2^{cn}]\}$.