Probability Theory and Related Fields

, Volume 81, Issue 1, pp 79–109 | Cite as

Brownian motion with a parabolic drift and airy functions

  • Piet Groeneboom


Let {W(t): t ∈ ∝} be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)=sup}t ∈ ∝: W(t)−(ta)2 is maximal}. Then {V(a): a ∈ ℝ} is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas fa(t)=(ta)2. We give an analytic expression for the infinitesimal generators of the processes a ∈ ℝ, in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in Groeneboom (1985), where the asymptotic distribution of the (standardized) L1-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process {W(t)−ct2: t ∈ ℝ}, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods.


Brownian Motion Transition Density Joint Density Airy Function Brownian Bridge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Nat. Bur. Stand. 55, Washington (1964)Google Scholar
  2. 2.
    Barbour, A.D.: Brownian motion and a sharply curved boundary. Adv. Appl. Probab. 13, 736–750 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blumenthal, R.M.: Weak convergence to Brownian excursion. Ann. Probab. 11, 798–800 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chernoff, H.: Estimation of the mode. Ann. Inst. Stat. Math. 16, 31–41 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Daniels, H.E.: The statistical theory of the strength of bundles of threads, I. Proc. Roy. Soc. Lond. Ser. A 183, 404–435 (1945)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Daniels, H.E.: The maximum size of a closed epidemic. Adv. Appl. Probab. 6, 607–621 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daniels, H.E., Skyrme, T.H.R.: The maximum of a random walk whose mean path has a maximum. Adv. Appl. Probab. 17, 85–99 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1969zbMATHGoogle Scholar
  9. 9.
    Doob, J.L.: Classical potential theory and its probabilistic counterpart. Berlin Heidelberg New York Tokyo: Springer 1984CrossRefzbMATHGoogle Scholar
  10. 10.
    Durrett, R. T., Iglehart, D. L., Miller, D. R.: Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5, 117–129 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grenander, U.: On the theory of mortality measurement. Part II. Skand. Akt. 39, 125–153 (1956)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Groeneboom, P.: The concave majorant of Brownian motion. Ann. Probab. 11, 1016–1027 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Groeneboom, P.: Estimating a monotone density. In: Le Cam, L.E., Olshen, R.A. (eds.) Proceedings of the Berkeley conferende in honor of Jerzy Neyman and Jack Kiefer, vol. II, pp. 539–555. Monterey: Wadsworth 1985Google Scholar
  14. 14.
    Houwelingen, J.C. van: Monotone empirical Bayes tests for uniform distributions using the maximum likelihood estimator of a decreasing density. Ann. Stat. 15, 875–879 (1987)CrossRefzbMATHGoogle Scholar
  15. 15.
    Itô, K., McKean, H.P. Jr.: Diffusion processes and their sample paths, 2nd ed. Berlin Heidelberg New York: Springer 1974zbMATHGoogle Scholar
  16. 16.
    Kiefer, J., Wolfowitz, J.: Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 73–85 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Louchard, G.: Kac's formula, Lévy's local time and Brownian excursion. J. Appl. Probab. 21, 479–499 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meyer, P.A., Smythe, R.T., Walsh, J.B.: Birth and death of Markov processes. Sixth Berkeley Symposium 3, 295–305. Berkeley: University of California Press 1972Google Scholar
  19. 19.
    Olver, F.W.J.: Asymptotics and special functions. New York: Academic Press 1974zbMATHGoogle Scholar
  20. 20.
    Phoenix, S.L., Taylor, H.M.: The asymptotic strength distribution of a general fiber bundle. Adv. Appl. Probab. 5, 200–216 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Prakasa Rao, B.L.S.: Estimation of a unimodal density. Sankhya Ser. A 31, 23–36 (1969)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Shepp, L.A.: On the integral of the absolute value of the pinned Wiener process. Ann. Probab. 10, 234–239 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Smith, R.L.: The asymptotic distribution of aseries-parallel system with equal load sharing. Ann. Probab. 10, 137–171 (1982)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979zbMATHGoogle Scholar
  25. 25.
    Temme, N.M.: A convolution integral equation solved by Laplace transformation. J. Comp. Appl. Math. 13, 609–613 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Venter, J.H.: On estimation of the mode. Ann. Math. Stat. 38, 1446–1456 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions. Proc. London Math. Soc. 28, 738–768 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Williams, D.: Diffusions, Markov processes and martingales. vol. 1: foundations. Chichester New York Brisbane Toronto: Wiley 1979zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Piet Groeneboom
    • 1
  1. 1.Department of MathematicsDelft University of TechnologyDelftThe Netherlands

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