Convex hulls of planar random walks with drift

TL;DR

This paper investigates the asymptotic behavior of the perimeter length of convex hulls formed by planar random walks with drift, establishing convergence of scaled variance and a central limit theorem.

Contribution

It proves the convergence of the scaled variance of the convex hull perimeter and derives a simple expression for its limit, addressing a question posed by Snyder and Steele.

Findings

Scaled variance of perimeter length converges to a non-zero limit.

A central limit theorem for the perimeter length is established in the non-degenerate case.

Provides a simple formula for the limiting variance of the perimeter length.

Abstract

Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $Var [ L_n] = O(n)$. We show that $n^{-1} Var [L_n]$ converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.