Electronic Communications in Probability

Convex minorants of random walks and Lévy processes

Josh Abramson, Jim Pitman, Nathan Ross, and Geronimo Uribe Bravo

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This article provides an overview of recent work on descriptions and properties of the Convex minorants of random walks and Lévy processes, which summarize and extend the literature on these subjects. The results surveyed include point process descriptions of the convex minorant of random walks and Lévy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander.

Article information

Electron. Commun. Probab. Volume 16 (2011), paper no. 38, 423-434.

Accepted: 19 August 2011
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G51: Processes with independent increments; Lévy processes

Random walks Lévy processes Brownian meander Convex minorant Uniform stick-breaking Fluctuation theory

This work is licensed under a Creative Commons Attribution 3.0 License.


Abramson, Josh; Pitman, Jim; Ross, Nathan; Uribe Bravo, Geronimo. Convex minorants of random walks and Lévy processes. Electron. Commun. Probab. 16 (2011), paper no. 38, 423--434. doi:10.1214/ECP.v16-1648. https://projecteuclid.org/euclid.ecp/1465261994

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