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Convex hulls of random walks and their scaling limits

Wade, Andrew R.; Xu, Chang

Convex hulls of random walks and their scaling limits Thumbnail


Authors

Chang Xu



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.

Citation

Wade, A. R., & Xu, C. (2015). Convex hulls of random walks and their scaling limits. Stochastic Processes and their Applications, 125(11), 4300-4320. https://doi.org/10.1016/j.spa.2015.06.008

Journal Article Type Article
Acceptance Date Jun 29, 2015
Online Publication Date Jul 9, 2015
Publication Date Nov 1, 2015
Deposit Date Aug 25, 2014
Publicly Available Date Sep 8, 2015
Journal Stochastic Processes and their Applications
Print ISSN 0304-4149
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 125
Issue 11
Pages 4300-4320
DOI https://doi.org/10.1016/j.spa.2015.06.008
Keywords Convex hull, Random walk, Brownian motion, Variance asymptotics, Scaling limits

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