Open Access
July 2012 The convex minorant of a Lévy process
Jim Pitman, Gerónimo Uribe Bravo
Ann. Probab. 40(4): 1636-1674 (July 2012). DOI: 10.1214/11-AOP658


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 $[0,1]$.


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Jim Pitman. Gerónimo Uribe Bravo. "The convex minorant of a Lévy process." Ann. Probab. 40 (4) 1636 - 1674, July 2012.


Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1248.60053
MathSciNet: MR2978134
Digital Object Identifier: 10.1214/11-AOP658

Primary: 60G51

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

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 4 • July 2012
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