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Convex hulls of planar random walks with drift


Authors: Andrew R. Wade and Chang Xu
Journal: Proc. Amer. Math. Soc. 143 (2015), 433-445
MSC (2010): Primary 60G50, 60D05; Secondary 60J10, 60F05
DOI: https://doi.org/10.1090/S0002-9939-2014-12239-8
Published electronically: September 16, 2014
MathSciNet review: 3272767
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Abstract: Denote by $ L_n$ the perimeter length of the convex hull of an $ n$-step planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $ n^{-1} L_n$ converges almost surely to a deterministic limit and proved an upper bound on the variance $ \mathbb{V}\mathrm {ar} [ L_n] = O(n)$. We show that $ n^{-1} \mathbb{V}\mathrm {ar} [L_n]$ 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 $ L_n$ in the non-degenerate case.


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Additional Information

Andrew R. Wade
Affiliation: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, United Kingdom
Email: andrew.wade@durham.ac.uk

Chang Xu
Affiliation: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
Email: c.xu@strath.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-12239-8
Received by editor(s): January 28, 2013
Received by editor(s) in revised form: April 18, 2013
Published electronically: September 16, 2014
Communicated by: David Levin
Article copyright: © Copyright 2014 American Mathematical Society