Electronic Journal of Probability

Vertices of the Least Concave Majorant of Brownian Motion with Parabolic Drift

Piet Groeneboom

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It was shown in Groeneboom (1983) that the least concave majorant of one-sided Brownian motion without drift can be characterized by a jump process with independent increments, which is the inverse of the process of slopes of the least concave majorant. This result can be used to prove the result in Sparre Andersen (1954) that the number of vertices of the smallest concave majorant of the empirical distribution function of a sample of size $n$ from the uniform distribution on $[0,1]$ is asymptotically normal, with an asymptotic expectation and variance which are both of order $\log(n)$. A similar (Markovian) inverse jump process was introduced in Groeneboom (1989), in an analysis of the least concave majorant of two-sided Brownian motion with a parabolic drift. This process is quite different from the process for one-sided Brownian motion without drift: the number of vertices in a (corresponding slopes) interval has an expectation proportional to the length of the interval and the variance of the number of vertices in such an interval is about half the size of the expectation, if the length of the interval tends to infinity. We prove an asymptotic normality result for the number of vertices in an increasing interval, which translates into a corresponding result for the least concave majorant of an empirical distribution function of a sample of size $n$, generated by a strictly concave distribution function. In this case the number of vertices is of order cube root $n$ and the variance is again about half the size of the asymptotic expectation. As a side result we obtain some interesting relations between the first moments of the number of vertices, the square of the location of the maximum of Brownian motion minus a parabola, the value of the maximum itself, the squared slope of the least concave majorant at zero, and the value of the least concave majorant at zero.

An erratum is available in EJP volume 18 paper 46.

Article information

Electron. J. Probab. Volume 16 (2011), paper no. 84, 2334-2358.

Accepted: 15 November 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 60J65: Brownian motion [See also 58J65]

Brownian motion parabolic drift number of vertices concave majorant Airy functions jump processes Grenander estimator

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


Groeneboom, Piet. Vertices of the Least Concave Majorant of Brownian Motion with Parabolic Drift. Electron. J. Probab. 16 (2011), paper no. 84, 2334--2358. doi:10.1214/EJP.v16-959. http://projecteuclid.org/euclid.ejp/1464820253.

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  • Behnen, Konrad. The Randles-Hogg test and an alternative proposal. Comm. Statist. 4 (1975), 203–238.
  • Chernoff, Herman. Estimation of the mode. Ann. Inst. Statist. Math. 16 1964 31–41.
  • Groeneboom, Piet. The concave majorant of Brownian motion. Ann. Probab. 11 (1983), no. 4, 1016–1027.
  • Groeneboom, P. Estimating a monotone density. Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, Vol. II (Berkeley, Calif., 1983), 539–555, Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, CA, 1985.
  • Groeneboom, Piet. Limit theorems for convex hulls. Probab. Theory Related Fields 79 (1988), no. 3, 327–368.
  • Groeneboom, Piet. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989), no. 1, 79–109.
  • Groeneboom, Piet. Convex hulls of uniform samples from a convex polygon}, To appear in the Advances in Applied Probability, 2011.
  • Groeneboom Piet, Hooghiemstra Gerard and Hendrik P. Lopuhaa, Asymptotic normality of the $L_1$ error of the Grenander estimator, Ann. Statist. 27 (1999), no. 4, 1316–1347.
  • Groeneboom, P.; Lopuhaä, H. P. Isotonic estimators of monotone densities and distribution functions: basic facts. Statist. Neerlandica 47 (1993), no. 3, 175–183.
  • Groeneboom, Piet; Pyke, Ronald. Asymptotic normality of statistics based on the convex minorants of empirical distribution functions. Ann. Probab. 11 (1983), no. 2, 328–345.
  • Groeneboom, Piet; Wellner, Jon A. Computing Chernoff's distribution. J. Comput. Graph. Statist. 10 (2001), no. 2, 388–400.
  • Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables.With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman.Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp.
  • Janson, Svante; Louchard, Guy; Martin-Löf, Anders. The maximum of Brownian motion with parabolic drift. Electron. J. Probab. 15 (2010), no. 61, 1893–1929.
  • Kurtz, Thomas G. Strong approximation theorems for density dependent Markov chains. Stochastic Processes Appl. 6 (1977/78), no. 3, 223–240.
  • Meyer, Mary; Woodroofe, Michael. On the degrees of freedom in shape-restricted regression. Ann. Statist. 28 (2000), no. 4, 1083–1104.
  • Nagaev, A. V. Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain. Ann. Inst. Statist. Math. 47 (1995), no. 1, 21–29.
  • 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.
  • Prakasa Rao, B. L. S. Estkmation of a unimodal density. Sankhya Ser. A 31 1969 23–36.
  • Scholz, F.-W. Combining independent $P$-values. A Festschrift for Erich L. Lehmann, pp. 379–394, Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, Calif., 1983.
  • Sparre Andersen, Erik. On the fluctuations of sums of random variables. II. Math. Scand. 2, (1954). 195–223.