Convex minorants of random walks and Lévy processes

Abstract

This article provides an overview of recent work on descriptions and properties of the convex minorant 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.

11 pages, 5 figures

References in corpus (3)

• The convex minorant of a Lévy process
• Bridges of Lévy processes conditioned to stay positive
• Concave Majorants of Random Walks and Related Poisson Processes

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• The Vervaat transform of Brownian bridges and Brownian motion
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• Dini derivatives for Exchangeable Increment processes and applications
• Asymptotic shape of the concave majorant of a Lévy process
• How long is the convex minorant of a one-dimensional random walk?
• Invariance principles for local times in regenerative settings
• Zeros of random tropical polynomials, random polytopes and stick-breaking
• Waiting Times for Ties in Random Competitions
• Lipschitz minorants of Brownian Motion and Levy processes
• Non-Markovianity of $2K-B$ and a degeneration