Talk by Jim Pitman (UC Berkeley)

Title: Concave majorants of random walks and Lévy processes

Abstract: Sparre Andersen and Spitzer in the 1950's related the structure of the concave majorants of random walks to the cycles of random permutations. Some aspects of this relation were extended to interval partitions associated with the concave majorant of Brownian motion and Brownian bridge by Suidan (2001) and by Balabdaoui and Pitman ( Other descriptions of concave majorants were obtained for Brownian motion by Groeneboom and Pitman in the early 1980's, and for Lévy processes by Nagasawa, Tanaka and Bertoin in the late 1990's. Recent joint work (with coauthors Josh Abramson, Nathan Ross, and Geronimo Uribe Bravo) fills out a complete array of combinatorial results for random walks and corresponding results for Brownian motion and Lévy processes. New results include simple explicit constructions of the concave majorant of a random walk on both finite and infinite time intervals, and of Poisson point processes of excursions between vertices of the concave majorant.

The talk will be based on the articles (, (, and (