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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convex hulls of random walks


Authors: Timothy Law Snyder and J. Michael Steele
Journal: Proc. Amer. Math. Soc. 117 (1993), 1165-1173
MSC: Primary 60D05; Secondary 60C05, 68Q25
MathSciNet review: 1169048
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Abstract: Features related to the perimeter of the convex hull $ {C_n}$ of a random walk in $ {\mathbb{R}^2}$ are studied, with particular attention given to its length $ {L_n}$. Bounds on the variance of $ {L_n}$ are obtained to show that, for walks with drift, $ {L_n}$ obeys a strong law. Exponential bounds on the tail probabilities of $ {L_n}$ under special conditions are also obtained. We then develop simple expressions for the expected values of other features of $ {C_n}$, including the number of faces, the sum of the lengths and squared lengths of the faces, and the number of faces of length $ t$ or less.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1169048-2
PII: S 0002-9939(1993)1169048-2
Keywords: Strong laws, convex hulls, random walks, Efron-Stein Inequality, variance bounds, geometric probability
Article copyright: © Copyright 1993 American Mathematical Society