Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: September 16, 2014
MathSciNet review: 3272767
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Glen Baxter, A combinatorial lemma for complex numbers, Ann. Math. Statist. 32 (1961), 901-904. MR 0126290 (23 #A3586)
  • [2] Richard Durrett, Probability: Theory and examples, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. MR 1068527 (91m:60002)
  • [3] Satya N. Majumdar, Alain Comtet, and Julien Randon-Furling, Random convex hulls and extreme value statistics, J. Stat. Phys. 138 (2010), no. 6, 955-1009. MR 2601420 (2011c:62166),
  • [4] Timothy Law Snyder and J. Michael Steele, Convex hulls of random walks, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1165-1173. MR 1169048 (93j:60013),
  • [5] F. Spitzer and H. Widom, The circumference of a convex polygon, Proc. Amer. Math. Soc. 12 (1961), 506-509. MR 0130616 (24 #A476)
  • [6] J. Michael Steele, The Bohnenblust-Spitzer algorithm and its applications: Probabilistic methods in combinatorics and combinatorial optimization, J. Comput. Appl. Math. 142 (2002), no. 1, 235-249. MR 1910531 (2003e:60027),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60G50, 60D05, 60J10, 60F05

Retrieve articles in all journals with MSC (2010): 60G50, 60D05, 60J10, 60F05

Additional Information

Andrew R. Wade
Affiliation: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, United Kingdom

Chang Xu
Affiliation: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

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

American Mathematical Society