Electronic Communications in Probability

Convex minorants of random walks and Lévy processes

Josh Abramson, Jim Pitman, Nathan Ross, and Geronimo Uribe Bravo

Full-text: Open access

Abstract

This article provides an overview of recent work on descriptions and properties of the Convex minorants 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.

Article information

Source
Electron. Commun. Probab. Volume 16 (2011), paper no. 38, 423-434.

Dates
Accepted: 19 August 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261994

Digital Object Identifier
doi:10.1214/ECP.v16-1648

Mathematical Reviews number (MathSciNet)
MR2831081

Zentralblatt MATH identifier
1243.60039

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G51: Processes with independent increments; Lévy processes

Keywords
Random walks Lévy processes Brownian meander Convex minorant Uniform stick-breaking Fluctuation theory

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Abramson, Josh; Pitman, Jim; Ross, Nathan; Uribe Bravo, Geronimo. Convex minorants of random walks and Lévy processes. Electron. Commun. Probab. 16 (2011), paper no. 38, 423--434. doi:10.1214/ECP.v16-1648. https://projecteuclid.org/euclid.ecp/1465261994


Export citation

References

  • Abramson, Josh; Pitman, Jim Concave majorants of random walks and related Poisson processes, ArXiv:1011.3262, (2010).
  • Sparre Andersen, Erik. On the fluctuations of sums of random variables. II. Math. Scand. 2, (1954). 195–223.
  • Bass, Richard F. Markov processes and convex minorants. Seminar on probability, XVIII, 29–41, Lecture Notes in Math., 1059, Springer, Berlin, 1984.
  • Bertoin, Jean. The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 (2000), 51–55 (electronic).
  • Brunk, H. D. A generalization of Spitzer's combinatorial lemma. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 1964 395–405 (1964).
  • Carolan, Chris; Dykstra, Richard. Characterization of the least concave majorant of Brownian motion, conditional on a vertex point, with application to construction. Ann. Inst. Statist. Math. 55 (2003), no. 3, 487–497.
  • Çinlar, Erhan. Sunset over Brownistan. Stochastic Process. Appl. 40 (1992), no. 1, 45–53.
  • Denisov, I. V. Random walk and the Wiener process considered from a maximum point. (Russian) Teor. Veroyatnost. i Primenen. 28 (1983), no. 4, 785–788.
  • Goldie, Charles M. Records, permutations and greatest convex minorants. Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 169–177.
  • Groeneboom, Piet. The concave majorant of Brownian motion. Ann. Probab. 11 (1983), no. 4, 1016–1027.
  • Lachièze-Rey, Raphaël, Concave Majorant of Stochastic Processes and Burgers Turbulence, Journal of Theoretical Probability, 2009.
  • Nagasawa, Masao. Stochastic processes in quantum physics. Monographs in Mathematics, 94. Birkhäuser Verlag, Basel, 2000. xviii+598 pp. ISBN: 3-7643-6208-1
  • Pitman, J. W. Remarks on the convex minorant of Brownian motion. Seminar on stochastic processes, 1982 (Evanston, Ill., 1982), 219–227, Progr. Probab. Statist., 5, Birkhäuser Boston, Boston, MA, 1983.
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Pitman, Jim; Ross, Nathan The convex minorant of Brownian motion, meander, and bridge, ArXiv:1011.3037, 2010.
  • Pitman, Jim; Uribe Bravo, Gerónimo The convex minorant of a Lévy process, ArXiv:1011.3069, 2010
  • Shepp, L. A.; Lloyd, S. P. Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 1966 340–357.
  • Spitzer, Frank. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 (1956), 323–339.
  • Suidan, T. M. Convex minorants of random walks and Brownian motion. Teor. Veroyatnost. i Primenen. 46 (2001), no. 3, 498–512; translation in Theory Probab. Appl. 46 (2003), no. 3, 469–481
  • Uribe Bravo, Gerónimo, Bridges of Lévy processes conditioned to stay positive, ArXiv:1101.4184, 2011.
  • Vervaat, Wim. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979), no. 1, 143–149.