Convex hulls of planar random walks with drift

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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. ISSN 0002-9939 (https://doi.org/10.1090/S0002-9939-2014-12239-8)

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Abstract

Denote by Ln the perimeter length of the convex hull of an n-step planar random walk whose increments have finite second moment and nonzero mean. Snyder and Steele showed that n-1Ln 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.

ORCID iDs

Wade, Andrew R. and Xu, Chang ORCID logoORCID: https://orcid.org/0000-0003-2674-0034;
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