## Electronic Journal of Probability

### How long is the convex minorant of a one-dimensional random walk?

#### Abstract

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.

#### Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 105, 22 pp.

Dates
Accepted: 23 July 2020
First available in Project Euclid: 5 September 2020

https://projecteuclid.org/euclid.ejp/1599271303

Digital Object Identifier
doi:10.1214/20-EJP497

#### Citation

Alsmeyer, Gerold; Kabluchko, Zakhar; Marynych, Alexander; Vysotsky, Vladislav. How long is the convex minorant of a one-dimensional random walk?. Electron. J. Probab. 25 (2020), paper no. 105, 22 pp. doi:10.1214/20-EJP497. https://projecteuclid.org/euclid.ejp/1599271303

#### References

• [1] Abramson, J. and Pitman, J. (2011). Concave majorants of random walks and related Poisson processes. Combinatorics, Probability and Computing 20(5), pp. 651–682.
• [2] Abramson, J., Pitman, J., Ross, N. and Uribe Bravo, G. (2011). Convex minorants of random walks and Lévy processes. Electron. Comm. Probab. 16, pp. 423–434.
• [3] Arratia, R., Barbour, A. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach, European Mathematical Society.
• [4] Arratia, R., Barbour, A. and Tavaré, S. (2006). A tale of three couplings: Poisson–Dirichlet and GEM approximations for random permutations. Combinatorics, Probability and Computing 15(1-2), pp. 31–62.
• [5] Billingsley, P. (1999). Convergence of Probability Measures, Second Ed., John Wiley & Sons.
• [6] Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation, Cambridge University Press.
• [7] Cai, G.-H. and Zhu, M.-H. (2006). A two-sided estimate in the Hsu–Robbins–Erdos law of large numbers for i.i.d. random variables sequence. J. Appl. Math. Comput. 22(1-2), pp. 331–338.
• [8] Davis, J. A. (1968). Convergence rates for the law of the iterated logarithm. Ann. Math. Stat. 39(5), pp. 1479–1485.
• [9] Erdos, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Statist. 20(2), pp. 286–291.
• [10] Friedman, N., Katz, M. and Koopman, L. H. (1966). Convergence rates for the central limit theorem. Proc. Nat. Acad. Sci. USA 56(4), pp. 1062–1065.
• [11] Gut, A. (2009). Stopped Random Walks, Second Ed., Springer.
• [12] Ibragimov, I. and Linnik, Yu. (1971). Independent and Stationary Sequences of Random Variables, Wolters–Noordhoff Publishing, Groningen.
• [13] Iksanov, A., Marynych, A. and Möhle, M. (2009). On the number of collisions in $beta(2,b)$-coalescents. Bernoulli 15(3), pp. 829–845.
• [14] Kingman, J. F. C. (1977). The population structure associated with the Ewens sampling formula. Theoretical Population Biology 11(2), pp. 274–283.
• [15] Lo, C. H., McRedmond, J. and Wallace, C. (2018). Functional limit theorems for random walks. arXiv:1810.06275.
• [16] McRedmond, J. (2019). Convex hulls of random walks. Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/13281/
• [17] McRedmond, J. and Wade, A. R. (2018). The convex hull of a planar random walk: perimeter, diameter, and shape. Electron. J. Probab. 23, paper no. 131.
• [18] Petrov, V. (1975). Sums of Independent Random Variables, Springer.
• [19] Pruss, A. R. (1997). A two-sided estimate in the Hsu–Robbins–Erdos law of large numbers. Stochastic Process. Appl. 70(2), pp. 173–180.
• [20] Sparre Andersen, E. (1954). On the fluctuations of sums of random variables II. Math. Scand. 2(2), pp. 195–223.
• [21] Vershik, A. M. and Shmidt, A. A. (1977). Limit measures arising in the asymptotic theory of symmetric groups. I. Theory of Probability and its Applications 22(1), pp. 70–85.
• [22] Wade, A. R. and Xu, C. (2015). Convex hulls of random walks and their scaling limits. Stochastic Process. Appl. 125(11), pp. 4300–4320.
• [23] Wade, A. R. and Xu, C. (2015). Convex hulls of planar random walks with drift. Proc. Amer. Math. Soc. 143(1), pp. 433–445.