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Remarks on the Convex Minorant of Brownian Motion

  • J. W. Pitman
Part of the Progress in Probability and Statistics book series (PRPR, volume 5)

Abstract

Recently Groeneboom [1] studied the concave majorant process of a Brownian motion (Bt, t ≤ O). The purpose of this note is to take a fresh look at some of Groeneboom’s results in the context of path decompositions of Williams [7], and to give a simple new description of this concave majorant process.

Keywords

Brownian Motion Time Inversion Poisson Point Process Independent Increment Bessel Process 
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References

  1. 1.
    P. Groeneboom. The concave majorant of Brownian motion. Technical Report No. 6, Dept. Statistics, Univ. of Washington, Seattle. To appear in Ann. Probab.Google Scholar
  2. 2.
    B. Maisonneuve. Exit systems. Ann. Probab. 3 (1975), 399–411.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    J.W. Pitman and M. Yor. Bessel processes and infinitely divisible laws. Stochastic Integrals, ed. D. Williams, 285–292. Lecture Notes in Math. 851. Springer-Verlag, Berlin, 1981.CrossRefGoogle Scholar
  4. 4.
    L.C.G. Rogers and J.W. Pitman. Markov functions. Ann. Probab. 9 (1981), 573–582.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S. Watanabe. On time inversion of one-dimensional diffusion processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31 (1975), 115–124.zbMATHCrossRefGoogle Scholar
  6. 6.
    D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 (1970), 871–873.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    D. Williams. Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc., Ser. 3, 28 (1974), 738–768.zbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser, Boston, Inc. 1983

Authors and Affiliations

  • J. W. Pitman
    • 1
  1. 1.Department of StatisticsUniv. of California at BerkeleyBerkeleyUSA

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