The convex hull of a planar random walk: perimeter, diameter, and shape

Abstract

We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n/D_n\to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then ${\mathrm{liminf}}_{n\to \infty }{L}_n/D_n=2$ and ${\mathrm{limsup}}_{n\to \infty }{L}_n/D_n=\pi$, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.

25 pages