Convex hulls of planar random walks with drift

Andrew R. Wade and Chang Xu

Proceedings of the American Mathematical Society, 143, no. 1, January 2015, 433–445. DOI: 10.1090/S0002-9939-2014-12239-8

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 $\mathrm{Var} \, L_n = O(n)$. We show that $n^{-1} \mathrm{Var} \, L_n$ 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.

Further remarks

Here is a picture of a convex hull of random walk with drift.