Convex hulls of planar random walks with drift
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- by Andrew R. Wade and Chang Xu
- Proc. Amer. Math. Soc. 143 (2015), 433-445
- DOI: https://doi.org/10.1090/S0002-9939-2014-12239-8
- Published electronically: September 16, 2014
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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.References
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Bibliographic 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
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3272767