Convex hulls of several multidimensional Gaussian random walks

TL;DR

This paper derives explicit formulas for the expected volume and facets of convex hulls formed by multiple multidimensional Gaussian random walks, extending known results for single walks and Gaussian polytopes.

Contribution

It provides new explicit formulas linking convex hull properties of Gaussian walks to Gaussian persistence probabilities, generalizing previous results.

Findings

Explicit formulas for expected volume and facets of convex hulls

Extension of known results to multiple Gaussian walks

Connections to Gaussian persistence probabilities

Abstract

We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the $d$-dimensional Gaussian polytope with or without the origin.