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.
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.
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|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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