The Annals of Probability

The convex minorant of a Lévy process

Jim Pitman and Gerónimo Uribe Bravo

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Abstract

We offer a unified approach to the theory of convex minorants of Lévy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Lévy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson–Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a Lévy process with continuous distributions above its convex minorant on the interval $[0,1]$.

Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1636-1674.

Dates
First available in Project Euclid: 4 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1341401145

Digital Object Identifier
doi:10.1214/11-AOP658

Mathematical Reviews number (MathSciNet)
MR2978134

Zentralblatt MATH identifier
1248.60053

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes

Keywords
Lévy processes convex minorant uniform stick-breaking fluctuation theory Vervaat transformation

Citation

Pitman, Jim; Uribe Bravo, Gerónimo. The convex minorant of a Lévy process. Ann. Probab. 40 (2012), no. 4, 1636--1674. doi:10.1214/11-AOP658. https://projecteuclid.org/euclid.aop/1341401145


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