Skip to main content
SpringerLink
Log in

Characterization of the least concave majorant of brownian motion, conditional on a vertex point, with application to construction

  • Characterization
  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Carolan, C. and Dykstra, R. (2001). Marginal distributions of the least concave majorant of Brownian motion,Ann. Statist.,29, 1732–1750.

    Article  MATH  MathSciNet  Google Scholar 

  • Groeneboom, P. (1983). The concave majorant of Brownian motion,Ann. Probab.,11, 1016–1027.

    MATH  MathSciNet  Google Scholar 

  • Itô, K. and McKean, H. P., jr. (1974).Diffusion Processes and Their Sample Paths, 2nd ed., Springer, Berlin.

    Google Scholar 

  • Pitman, J. W. (1983). Remarks on the convex minorant of Brownian motion,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor). 219–228, Birkhäuser Boston.

    Google Scholar 

  • Williams, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions, I,Proc. London Math. Soc. (3),28, 738–768.

    MATH  MathSciNet  Google Scholar 

  • Woodroofe, M. and Sun, J. (1999). Testing uniformity versus a monotone density,Ann. Statist.,27, 338–360.

    Article  MATH  MathSciNet  Google Scholar 

  • Wu, W. B., Woodroofe, M. and Mentz, G. (2001). Isotonic regression: Another look at the changepoint problem,Biometrika,88, 793–804.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, East Carolina University, 129 Austin, 27858, Greenville, NC, U.S.A.

    Chris Carolan

  2. Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer, 52242, Iowa City, IA, U.S.A.

    Richard Dykstra

Authors
  1. Chris Carolan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard Dykstra
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Carolan, C., Dykstra, R. Characterization of the least concave majorant of brownian motion, conditional on a vertex point, with application to construction. Ann Inst Stat Math 55, 487–497 (2003). https://doi.org/10.1007/BF02517802

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02517802

Key words and phrases

Search

Navigation