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

McRedmond, James; Wade, Andrew R.

doi:10.1214/18-ejp257

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

McRedmond, James; Wade, Andrew R.

Authors

James McRedmond

Professor Andrew Wade andrew.wade@durham.ac.uk
Professor

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 Ln, diameter Dn, and shape. In the case where the walk has a non-zero mean drift, we show that Ln=Dn ! 2 a.s., and give distributional limit theorems and variance asymptotics for Dn, 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 lim infn!1 Ln=Dn = 2 and lim supn!1 Ln=Dn = , a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.

Citation

McRedmond, J., & Wade, A. R. (2018). The convex hull of a planar random walk: perimeter, diameter, and shape. Electronic Journal of Probability, 23, Article 131. https://doi.org/10.1214/18-ejp257

Journal Article Type	Article
Acceptance Date	Dec 12, 2018
Online Publication Date	Dec 22, 2018
Publication Date	Dec 22, 2018
Deposit Date	Mar 23, 2018
Publicly Available Date	Dec 27, 2018
Journal	Electronic Journal of Probability
Publisher	Institute of Mathematical Statistics
Peer Reviewed	Peer Reviewed
Volume	23
Article Number	131
DOI	https://doi.org/10.1214/18-ejp257