Electronic Journal of Probability

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

Gerold Alsmeyer, Zakhar Kabluchko, Alexander Marynych, and Vladislav Vysotsky

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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
Received: 26 September 2019
Accepted: 23 July 2020
First available in Project Euclid: 5 September 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1599271303

Digital Object Identifier
doi:10.1214/20-EJP497

Subjects
Primary: 60F05: Central limit and other weak theorems 60G55: Point processes
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
convex minorant random permutation random walk

Rights
Creative Commons Attribution 4.0 International License.

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


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