Probability Theory and Related Fields

, Volume 138, Issue 1–2, pp 269–304 | Cite as

Weak convergence of the scaled median of independent Brownian motions

Article

Abstract

We consider the median of n independent Brownian motions, denoted by Mn(t), and show that \(\sqrt{n}\,M_n\) converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be Hölder continuous with exponent γ for all γ < 1/4.

Keywords

Brownian motion Median Weak convergence Fractional Brownian motion Tightness 

Mathematics Subject Classification

60F17 60G15 60J65 60K35 

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References

  1. 1.
    Dürr D., Goldstein S., Lebowitz J.L. (1985) Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Commun. Pure Appl. Math. 38, 575–597Google Scholar
  2. 2.
    Harris T.E. (1965) Diffusions with collisions between particles. J. Appl. Probab. 2, 323–338CrossRefGoogle Scholar
  3. 3.
    Karatzas I., Shreve S.E. (1991) Brownian Motion and Stochastic Calculus. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  4. 4.
    Reiss R.D. (1989) Approximate Distributions of Order Statistics With Applications to Nonparametric Statistics. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  5. 5.
    Spitzer F. (1968) Uniform motion with elastic collisions of an infinite particle system. J. Math. Mech. 18, 973–989MathSciNetGoogle Scholar
  6. 6.
    Stroock D.W. (1993) Probability Theory, An Analytic View. Cambridge University Press, CambridgeMATHGoogle Scholar
  7. 7.
    Swanson J. (2004) Variations of Stochastic Processes. Alternative Approaches. Doctoral Dissertation, University of WashingtonGoogle Scholar
  8. 8.
    Tupper P.F. (2005) A test problem for molecular dynamics integrators. IMA J. Numer. Anal. 25(2): 286–309MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Wisconsin-MadisonMadisonUSA

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