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Convex hulls of random walks and their scaling limits.

Wade, Andrew R. and Xu, Chang (2015) 'Convex hulls of random walks and their scaling limits.', Stochastic processes and their applications., 125 (11). pp. 4300-4320.

Abstract

For the perimeter length and the area of the convex hull of the first n steps of a planar random walk, we study n→∞ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero.

Item Type:Article
Keywords:Convex hull, Random walk, Brownian motion, Variance asymptotics, Scaling limits
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Available under License - Creative Commons Attribution.
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1016/j.spa.2015.06.008
Publisher statement:© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Date accepted:29 June 2015
Date deposited:08 September 2015
Date of first online publication:09 July 2015
Date first made open access:No date available

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