The Annals of Probability

The convex minorant of a Lévy process

Jim Pitman and Gerónimo Uribe Bravo

Full-text: Open access


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 $[0,1]$.

Article information

Ann. Probab., Volume 40, Number 4 (2012), 1636-1674.

First available in Project Euclid: 4 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes

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


Pitman, Jim; Uribe Bravo, Gerónimo. The convex minorant of a Lévy process. Ann. Probab. 40 (2012), no. 4, 1636--1674. doi:10.1214/11-AOP658.

Export citation


  • Abramson, J. and Pitman, J. (2011). Concave majorants of random walks and related Poisson processes. Combinatorics Probab. Comput. 20 651–682.
  • Abramson, J., Pitman, J., Ross, N. and Uribe Bravo, G. (2011). Convex minorants of random walks and Lévy processes. Electron. Commun. Probab. 16 423–434.
  • Andersen, E. S. (1950). On the frequency of positive partial sums of a series of random variables. Mat. Tidsskr. B 1950 33–35.
  • Andersen, E. S. (1953a). On sums of symmetrically dependent random variables. Skand. Aktuarietidskr. 36 123–138.
  • Andersen, E. S. (1953b). On the fluctuations of sums of random variables. Math. Scand. 1 263–285.
  • Andersen, E. S. (1954). On the fluctuations of sums of random variables. II. Math. Scand. 2 195–223.
  • Balabdaoui, F. and Pitman, J. (2009). The distribution of the maximal difference between Brownian bridge and its concave majorant. Bernoulli 17 466–483.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • Bertoin, J. (2000). The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 51–55 (electronic).
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Chaumont, L. (1997). Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 377–403.
  • Chaumont, L. (2010). On the law of the supremum of Lévy processes. Available at
  • Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948–961 (electronic).
  • Chung, K. L. and Fuchs, W. H. J. (1951). On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 1951 12.
  • Chung, K. L. and Ornstein, D. (1962). On the recurrence of sums of random variables. Bull. Amer. Math. Soc. 68 30–32.
  • Doney, R. A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin.
  • Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185 371–381.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Fourati, S. (2005). Vervaat et Lévy. Ann. Inst. H. Poincaré Probab. Statist. 41 461–478.
  • Gihman, Ĭ. Ī. and Skorohod, A. V. (1975). The Theory of Stochastic Processes. II. Springer, New York.
  • Greenwood, P. and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. in Appl. Probab. 12 893–902.
  • Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Knight, F. B. (1996). The uniform law for exchangeable and Lévy process bridges. Astérisque 236 171–188.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Lachieze-Rey, R. (2009). Concave majorant of stochastic processes and Burgers turbulence. J. Theor. Probab. To appear. DOI:10.1007/S10959-011-0354-7.
  • McCloskey, J. W. (1965). A model for the distribution of individuals by species in an environment. Ph.D. thesis, Michigan State Univ., Ann Anbor, MI.
  • Miermont, G. (2001). Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps. Electron. J. Probab. 6 33 pp. (electronic).
  • Millar, P. W. (1977). Zero–one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226 365–391.
  • Nagasawa, M. (2000). Stochastic Processes in Quantum Physics. Monographs in Mathematics 94 355–388. Birkhäuser, Basel.
  • Pečerskiĭ, E. A. and Rogozin, B. A. (1969). The combined distributions of the random variables connected with the fluctuations of a process with independent increments. Teor. Veroyatn. Primen. 14 431–444.
  • Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21–39.
  • Pitman, J. W. (1983). Remarks on the convex minorant of Brownian motion. In Seminar on Stochastic Processes, 1982 (Evanston, IL, 1982). Progress in Probability Statist. 5 219–227. Birkhäuser, Boston, MA.
  • Pitman, J. and Ross, N. (2010). The greatest convex minorant of Brownian motion, meander, and bridge. Probab. Theor. Related Fields. To appear. DOI:10.1007/S00440-011-0385-0.
  • Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855–900.
  • Rogozin, B. A. (1968). The local behavior of processes with independent increments. Teor. Veroyatn. Primen. 13 507–512.
  • Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Stud. Adv. Math. 68. Cambridge Univ. Press, Cambridge.
  • Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 323–339.
  • Suidan, T. M. (2001a). Convex minorants of random walks and Brownian motion. Teor. Veroyatn. Primen. 46 498–512.
  • Suidan, T. M. (2001b). A one-dimensional gravitationally interacting gas and the convex minorant of Brownian motion. Uspekhi Mat. Nauk 56 73–96.
  • Uribe Bravo, G. (2011). Bridges of Lévy processes conditioned to stay positive. Available at
  • Vervaat, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143–149.
  • Vigon, V. (2002). Votre Lévy rampe-t-il? J. Lond. Math. Soc. (2) 65 243–256.