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

McRedmond, James; Wade, Andrew R.

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
• Electronic ISSN: 1083-6489
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 23
• Article Number: 131
• DOI: https://doi.org/10.1214/18-ejp257
• Public URL: https://durham-repository.worktribe.com/output/1336550