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

  • Jason SwansonEmail author


We consider the median of n independent Brownian motions, denoted by M n (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.


Brownian motion Median Weak convergence Fractional Brownian motion Tightness 

Mathematics Subject Classification

60F17 60G15 60J65 60K35 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Wisconsin-MadisonMadisonUSA

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