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

McRedmond, James and Wade, Andrew R. (2018) 'The convex hull of a planar random walk : perimeter, diameter, and shape.', Electronic journal of probability., 23 . p. 131.

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.

Item Type:Article
Full text:Publisher-imposed embargo
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1214/18-EJP257
Publisher statement:This article has been published under a Creative Commons CC BY 4.0. licence.
Date accepted:12 December 2018
Date deposited:13 December 2018
Date of first online publication:22 December 2018
Date first made open access:27 December 2018

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