Convex hulls of planar random walks with drift

Wade, Andrew R.; Xu, Chang

Abstract

Denote by Ln the length of the perimeter of the convex hull of n steps of a planar random walk whose increments have nite second moment and non-zero mean. Snyder and Steele showed that -1 Ln converges almost surely to a deterministic limit, and proved an upper bound on the variance Var[Ln] = O(n). We show that n-1Var[Ln] converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for Ln in the non-degenerate case.

Citation

Wade, A. R., & Xu, C. (2015). Convex hulls of planar random walks with drift. Proceedings of the American Mathematical Society, 143(1), 433-445. https://doi.org/10.1090/s0002-9939-2014-12239-8

• Journal Article Type: Article
• Acceptance Date: Apr 18, 2013
• Online Publication Date: Sep 16, 2014
• Publication Date: Jan 1, 2015
• Deposit Date: May 15, 2013
• Publicly Available Date: Jul 12, 2013
• Journal: Proceedings of the American Mathematical Society
• Print ISSN: 0002-9939
• Electronic ISSN: 1088-6826
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 143
• Issue: 1
• Pages: 433-445
• DOI: https://doi.org/10.1090/s0002-9939-2014-12239-8
• Keywords: Convex hull, Random walk, Variance asymptotics, Central limit theorem.
• Public URL: https://durham-repository.worktribe.com/output/1456146