Convex hulls of random walks and their scaling limits

Wade, Andrew R.; Xu, Chang

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
• Electronic ISSN: 1879-209X
• 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
• Public URL: https://durham-repository.worktribe.com/output/1455155