## Electronic Journal of Probability

### 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 $\liminf _{n \to \infty } L_n/D_n =2$ and $\limsup _{n \to \infty } L_n /D_n = \pi$, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 131, 24 pp.

Dates
Accepted: 12 December 2018
First available in Project Euclid: 22 December 2018

https://projecteuclid.org/euclid.ejp/1545447916

Digital Object Identifier
doi:10.1214/18-EJP257

Mathematical Reviews number (MathSciNet)
MR3896868

Zentralblatt MATH identifier
07021687

#### Citation

McRedmond, James; Wade, Andrew R. The convex hull of a planar random walk: perimeter, diameter, and shape. Electron. J. Probab. 23 (2018), paper no. 131, 24 pp. doi:10.1214/18-EJP257. https://projecteuclid.org/euclid.ejp/1545447916

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