Convex hulls of stable random walks

TL;DR

This paper studies the asymptotic behavior of convex hulls formed by stable random walks in multi-dimensional space, showing they converge to the convex hull of a stable Lévy process, with applications to intrinsic volumes.

Contribution

It proves the convergence of convex hulls of stable random walks to those of a stable Lévy process and establishes convergence of expected intrinsic volumes.

Findings

Convex hulls of stable random walks converge to the hull of a stable Lévy process.

Expected intrinsic volumes of the convex hulls also converge under mild conditions.

The results extend understanding of geometric properties of stable processes in high dimensions.

Abstract

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable L\'{e}vy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.