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.
References
[1] Bhattacharya, R. N. and Rao, R. R.: Normal Approximation and Asymptotic Expansions. Updated reprint of the 1986 edition, SIAM, Philadelphia, 2010.[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. MR1700749 0944.60003[2] Billingsley, P.: Convergence of Probability Measures. 2nd ed., Wiley, New York, 1999. MR1700749 0944.60003
[3] Billingsley, P.: Probability and Measure. Anniversary ed., Wiley, New York, 2012. 1236.60001[3] Billingsley, P.: Probability and Measure. Anniversary ed., Wiley, New York, 2012. 1236.60001
[4] Durrett, R.: Probability: Theory and Examples. 4th ed., Cambridge University Press, Cambridge, 2010. 1202.60001[4] Durrett, R.: Probability: Theory and Examples. 4th ed., Cambridge University Press, Cambridge, 2010. 1202.60001
[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. MR3161524 10.1214/12-AIHP515 euclid.aihp/1388545267[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. MR3161524 10.1214/12-AIHP515 euclid.aihp/1388545267
[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.[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. 1139.52001[8] Gruber, P. M.: Convex and Discrete Geometry. Springer, Berlin, 2007. 1139.52001
[9] Gut, A.: Probability: A Graduate Course. Springer, Berlin, 2005. 1076.60001[9] Gut, A.: Probability: A Graduate Course. Springer, Berlin, 2005. 1076.60001
[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. 1373.52009 10.1007/s00039-017-0415-x[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. 1373.52009 10.1007/s00039-017-0415-x
[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. 1377.52007 10.1016/j.aim.2017.09.002[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. 1377.52007 10.1016/j.aim.2017.09.002
[12] Kallenberg, O.: Foundations of Modern Probability. 2nd ed., Springer, 2002. 0996.60001[12] Kallenberg, O.: Foundations of Modern Probability. 2nd ed., Springer, 2002. 0996.60001
[13] Lo, C. H., McRedmond, J. and Wallace, C.: Functional limit theorems for random walks. arXiv:1810.06275 1810.06275[13] Lo, C. H., McRedmond, J. and Wallace, C.: Functional limit theorems for random walks. arXiv:1810.06275 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. 1188.82024 10.1007/s10955-009-9905-z[14] Majumdar, S. N., Comtet, A. and Randon-Furling, J.: Random convex hulls and extreme value statistics. J. Stat. Phys. 138 (2010) 955–1009. 1188.82024 10.1007/s10955-009-9905-z
[15] McRedmond, J. and Xu, C.: On the expected diameter of planar Brownian motion. Statist. Probab. Lett. 130 (2017) 1–4. 1386.60284 10.1016/j.spl.2017.07.001[15] McRedmond, J. and Xu, C.: On the expected diameter of planar Brownian motion. Statist. Probab. Lett. 130 (2017) 1–4. 1386.60284 10.1016/j.spl.2017.07.001
[16] Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford, 2003. 1029.60007[16] Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford, 2003. 1029.60007
[17] Rudnick, J. and Gaspari, G.: The shapes of random walks. Science 236 (1987) 384–389. 0625.60080 10.1088/0305-4470/20/4/031[17] Rudnick, J. and Gaspari, G.: The shapes of random walks. Science 236 (1987) 384–389. 0625.60080 10.1088/0305-4470/20/4/031
[18] Snyder, T. L. and Steele, J. M.: Convex hulls of random walks. Proc. Amer. Math. Soc. 117 (1993) 1165–1173. 0770.60011 10.1090/S0002-9939-1993-1169048-2[18] Snyder, T. L. and Steele, J. M.: Convex hulls of random walks. Proc. Amer. Math. Soc. 117 (1993) 1165–1173. 0770.60011 10.1090/S0002-9939-1993-1169048-2
[19] Spitzer, F. and Widom, H.: The circumference of a convex polygon. Proc. Amer. Math. Soc. 12 (1961) 506–509. 0099.38501 10.1090/S0002-9939-1961-0130616-7[19] Spitzer, F. and Widom, H.: The circumference of a convex polygon. Proc. Amer. Math. Soc. 12 (1961) 506–509. 0099.38501 10.1090/S0002-9939-1961-0130616-7
[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. 1377.52008 10.1214/15-AOP1079 euclid.aop/1490947312[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. 1377.52008 10.1214/15-AOP1079 euclid.aop/1490947312
[21] Vysotsky, V. and Zaporozhets, D.: Convex hulls of multidimensional random walks. Trans. Amer. Math. Soc. 370 (2018) 7985–8012. 06935861 10.1090/tran/7253[21] Vysotsky, V. and Zaporozhets, D.: Convex hulls of multidimensional random walks. Trans. Amer. Math. Soc. 370 (2018) 7985–8012. 06935861 10.1090/tran/7253
[22] Wade, A. R. and Xu, C.: Convex hulls of planar random walks with drift. Proc. Amer. Math. Soc. 143 (2015) 433–445. MR3272767 10.1090/S0002-9939-2014-12239-8[22] Wade, A. R. and Xu, C.: Convex hulls of planar random walks with drift. Proc. Amer. Math. Soc. 143 (2015) 433–445. MR3272767 10.1090/S0002-9939-2014-12239-8
[23] Wade, A. R. and Xu, C.: Convex hulls of random walks and their scaling limits. Stochastic Process. Appl. 125 (2015), 4300–4320. 1322.60052 10.1016/j.spa.2015.06.008[23] Wade, A. R. and Xu, C.: Convex hulls of random walks and their scaling limits. Stochastic Process. Appl. 125 (2015), 4300–4320. 1322.60052 10.1016/j.spa.2015.06.008
[24] Xu, C.: Convex Hulls of Planar Random Walks. PhD thesis, University of Strathclyde, 2017. arXiv:1704.01377 1704.01377[24] Xu, C.: Convex Hulls of Planar Random Walks. PhD thesis, University of Strathclyde, 2017. arXiv:1704.01377 1704.01377
