Electronic Journal of Probability

The convex hull of a planar random walk: perimeter, diameter, and shape

James McRedmond and Andrew R. Wade

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We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf _{n \to \infty } L_n/D_n =2$ and $\limsup _{n \to \infty } L_n /D_n = \pi $, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 131, 24 pp.

Received: 23 March 2018
Accepted: 12 December 2018
First available in Project Euclid: 22 December 2018

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F20: Zero-one laws

random walk convex hull perimeter length diameter shape zero-one law

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McRedmond, James; Wade, Andrew R. The convex hull of a planar random walk: perimeter, diameter, and shape. Electron. J. Probab. 23 (2018), paper no. 131, 24 pp. doi:10.1214/18-EJP257. https://projecteuclid.org/euclid.ejp/1545447916

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  • [1] Bhattacharya, R. N. and Rao, R. R.: Normal Approximation and Asymptotic Expansions. Updated reprint of the 1986 edition, SIAM, Philadelphia, 2010.
  • [2] Billingsley, P.: Convergence of Probability Measures. 2nd ed., Wiley, New York, 1999.
  • [3] Billingsley, P.: Probability and Measure. Anniversary ed., Wiley, New York, 2012.
  • [4] Durrett, R.: Probability: Theory and Examples. 4th ed., Cambridge University Press, Cambridge, 2010.
  • [5] Eldan, R.: Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 95–110.
  • [6] Eldan, R.: Volumetric properties of the convex hull of an $n$-dimensional Brownian motion. Electron. J. Probab. 19 (2014) paper no. 45.
  • [7] Grebenkov, D. S., Lanoiselée, Y. and Majumdar, S. N.: Mean perimeter and mean area of the convex hull over planar random walks. J. Statist. Mech. Theor. Exp. (2017) 103203.
  • [8] Gruber, P. M.: Convex and Discrete Geometry. Springer, Berlin, 2007.
  • [9] Gut, A.: Probability: A Graduate Course. Springer, Berlin, 2005.
  • [10] Kabluchko, Z., Vysotsky, V. and Zaporozhets, D.: Convex hulls of random walks, hyperplane arrangements, and Weyl chambers. Geom. Funct. Anal. 27 (2017) 880–918.
  • [11] Kabluchko, Z., Vysotsky, V. and Zaporozhets, D.: Convex hulls of random walks: Expected number of faces and face probabilities. Adv. Math. 320 (2017) 595–629.
  • [12] Kallenberg, O.: Foundations of Modern Probability. 2nd ed., Springer, 2002.
  • [13] Lo, C. H., McRedmond, J. and Wallace, C.: Functional limit theorems for random walks. arXiv:1810.06275
  • [14] Majumdar, S. N., Comtet, A. and Randon-Furling, J.: Random convex hulls and extreme value statistics. J. Stat. Phys. 138 (2010) 955–1009.
  • [15] McRedmond, J. and Xu, C.: On the expected diameter of planar Brownian motion. Statist. Probab. Lett. 130 (2017) 1–4.
  • [16] Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford, 2003.
  • [17] Rudnick, J. and Gaspari, G.: The shapes of random walks. Science 236 (1987) 384–389.
  • [18] Snyder, T. L. and Steele, J. M.: Convex hulls of random walks. Proc. Amer. Math. Soc. 117 (1993) 1165–1173.
  • [19] Spitzer, F. and Widom, H.: The circumference of a convex polygon. Proc. Amer. Math. Soc. 12 (1961) 506–509.
  • [20] Tikhomirov, K. and Youssef, P.: When does a discrete-time random walk in $\mathbb{R} ^n$ absorb the origin into its convex hull? Ann. Probab. 45 (2017) 965–1002.
  • [21] Vysotsky, V. and Zaporozhets, D.: Convex hulls of multidimensional random walks. Trans. Amer. Math. Soc. 370 (2018) 7985–8012.
  • [22] Wade, A. R. and Xu, C.: Convex hulls of planar random walks with drift. Proc. Amer. Math. Soc. 143 (2015) 433–445.
  • [23] Wade, A. R. and Xu, C.: Convex hulls of random walks and their scaling limits. Stochastic Process. Appl. 125 (2015), 4300–4320.
  • [24] Xu, C.: Convex Hulls of Planar Random Walks. PhD thesis, University of Strathclyde, 2017. arXiv:1704.01377