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Convex hulls of planar random walks with drift.

Wade, Andrew R. and Xu, Chang (2015) 'Convex hulls of planar random walks with drift.', Proceedings of the American Mathematical Society., 143 (1). pp. 433-445.

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.

Item Type:Article
Keywords:Convex hull, Random walk, Variance asymptotics, Central limit theorem.
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1090/S0002-9939-2014-12239-8
Publisher statement:First published in Proceedings of the American Mathematical Society in 143 (1), 2015, published by the American Mathematical Society.
Date accepted:18 April 2013
Date deposited:12 July 2013
Date of first online publication:16 September 2014
Date first made open access:No date available

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