Weak Convergence of the Scaled Median of Independent Brownian Motions

Details Weak Convergence of the Scaled Median of Independent Brownian Motions Weak Convergence of the Scaled Median of Independent Brownian Motions

2005-07-26

arxiv.org/abs/math/0507524v2

Probab. Theory Related Fields, 138 (2007), Nos. 1-2, 269--304

Mathematics

Probability

to appear in Probability Theory and Related Fields

Scientific paper

10.1007/s00440-006-0024-3

We consider the median of n independent Brownian motions, and show that this process, when properly scaled, 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 Holder continuous with exponent gamma for all gamma < 1/4.

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