Convex hulls of random walks and their scaling limits

TL;DR

This paper investigates the asymptotic behavior of the perimeter and area of convex hulls formed by planar random walks, revealing non-Gaussian limits and scaling behaviors under mild conditions.

Contribution

It establishes non-Gaussian distributional limits for convex hull metrics of random walks, extending understanding beyond classical Gaussian results.

Findings

Perimeter length with drift satisfies a CLT.

Non-zero limiting variances confirmed.

Scaling limits linked to Brownian motion convex hulls.

Abstract

For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study $n \to \infty$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero.