Weak convergence of the scaled median of independent Brownian motions |
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Authors: | Jason Swanson |
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Institution: | (1) Mathematics Department, University of Wisconsin-Madison, Madison, WI, USA |
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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_nWe consider the median of n independent Brownian motions, denoted by M
n
(t), and show that
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 H?lder continuous with exponent γ for all γ < 1/4.
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Keywords: | Brownian motion Median Weak convergence Fractional Brownian motion Tightness |
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