The convex minorant of a Lévy process
Jim Pitman and Gerónimo Uribe Bravo
Source: Ann. Probab. Volume 40, Number 4
(2012), 1636-1674.
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 $[0,1]$.
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60G51
Keywords: Lévy processes; convex minorant; uniform stick-breaking; fluctuation theory; Vervaat transformation
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