Weak convergence of the scaled median of independent Brownian motions

@article{Swanson2007WeakCO,
  title={Weak convergence of the scaled median of independent Brownian motions},
  author={Jason Swanson},
  journal={Probability Theory and Related Fields},
  year={2007},
  volume={138},
  pages={269-304}
}
  • Jason Swanson
  • Published 26 July 2005
  • Mathematics
  • Probability Theory and Related Fields
AbstractWe 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. 
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