Convex minorants of random walks and Lévy processes

Josh Abramson (University of California, Berkeley)
Jim Pitman (University of California, Berkeley)
Nathan Ross (University of California, Berkeley)
Geronimo Uribe Bravo (Universidad Nacional Autónoma de México)

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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 423-434

Publication Date: August 19, 2011

DOI: 10.1214/ECP.v16-1648

References

  1. Abramson, Josh; Pitman, Jim Concave majorants of random walks and related Poisson processes, ArXiv:1011.3262, (2010).
  2. Sparre Andersen, Erik. On the fluctuations of sums of random variables. II. Math. Scand. 2, (1954). 195--223. MR0068154 (16,839e)
  3. Bass, Richard F. Markov processes and convex minorants. Seminar on probability, XVIII, 29--41, Lecture Notes in Math., 1059, Springer, Berlin, 1984. MR0770946 (86d:60086)
  4. Bertoin, Jean. The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 (2000), 51--55 (electronic). MR1747095 (2001i:60081)
  5. Brunk, H. D. A generalization of Spitzer's combinatorial lemma. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 1964 395--405 (1964). MR0162302 (28 #5501)
  6. 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. MR2007793 (2004h:60123)
  7. Çinlar, Erhan. Sunset over Brownistan. Stochastic Process. Appl. 40 (1992), no. 1, 45--53. MR1145458 (93f:60120)
  8. 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. MR0726906 (85f:60117)
  9. Goldie, Charles M. Records, permutations and greatest convex minorants. Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 169--177. MR0994088 (91a:60138)
  10. Groeneboom, Piet. The concave majorant of Brownian motion. Ann. Probab. 11 (1983), no. 4, 1016--1027. MR0714964 (85h:60119)
  11. Lachièze-Rey, Raphaël, Concave Majorant of Stochastic Processes and Burgers Turbulence, Journal of Theoretical Probability, 2009 .
  12. Nagasawa, Masao. Stochastic processes in quantum physics. Monographs in Mathematics, 94. Birkhäuser Verlag, Basel, 2000. xviii+598 pp. ISBN: 3-7643-6208-1 MR1739699 (2001g:60148)
  13. 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. MR0733673 (85f:60119)
  14. 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 MR2245368 (2008c:60001)
  15. Pitman, Jim; Ross, Nathan The convex minorant of Brownian motion, meander, and bridge, ArXiv:1011.3037 , 2010.
  16. Pitman, Jim; Uribe Bravo, Gerónimo The convex minorant of a Lévy process, ArXiv:1011.3069 , 2010
  17. Shepp, L. A.; Lloyd, S. P. Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 1966 340--357. MR0195117 (33 #3320)
  18. Spitzer, Frank. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 (1956), 323--339. MR0079851 (18,156e)
  19. 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 MR1978665 (2004d:60097)
  20. Uribe Bravo, Gerónimo, Bridges of Lévy processes conditioned to stay positive, ArXiv:1101.4184, 2011.
  21. Vervaat, Wim. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979), no. 1, 143--149. MR0515820 (80b:60107)
Bookmark and Share


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