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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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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DOI: https://doi.org/10.1007/s00440-006-0024-3
Keywords
- Brownian motion
- Median
- Weak convergence
- Fractional Brownian motion
- Tightness
Mathematics Subject Classification
- 60F17
- 60G15
- 60J65
- 60K35