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Weak convergence of the scaled median of independent Brownian motions
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  • Published: 18 August 2006

Weak convergence of the scaled median of independent Brownian motions

  • Jason Swanson1 

Probability Theory and Related Fields volume 138, pages 269–304 (2007)Cite this article

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  • 12 Citations

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Abstract

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.

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Authors and Affiliations

  1. Mathematics Department, University of Wisconsin-Madison, Madison, WI, USA

    Jason Swanson

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  1. Jason Swanson
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Correspondence to Jason Swanson.

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Cite this article

Swanson, J. Weak convergence of the scaled median of independent Brownian motions. Probab. Theory Relat. Fields 138, 269–304 (2007). https://doi.org/10.1007/s00440-006-0024-3

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  • Received: 28 July 2005

  • Revised: 25 June 2006

  • Published: 18 August 2006

  • Issue Date: May 2007

  • DOI: https://doi.org/10.1007/s00440-006-0024-3

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Keywords

  • Brownian motion
  • Median
  • Weak convergence
  • Fractional Brownian motion
  • Tightness

Mathematics Subject Classification

  • 60F17
  • 60G15
  • 60J65
  • 60K35
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