Convex hulls of planar random walks with drift

Wade, Andrew R.; Xu, Chang

doi:10.1090/s0002-9939-2014-12239-8

Convex hulls of planar random walks with drift

Wade, Andrew R.; Xu, Chang

Authors

Professor Andrew Wade andrew.wade@durham.ac.uk
Professor

Chang Xu

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.

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Copyright Statement
First published in Proceedings of the American Mathematical Society in 143 (1), 2015, published by the American Mathematical Society.