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Empirical Processes and p-variation

  • R. M. Dudley
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Remainder bounds in Fréchet differentiability of functionals for p-variation norms are found for empirical distribution functions. For \(1 \le p \le 2\) the p-variation of the empirical process \(n^{1/2} (F_n - F)\) is of order \(n^{1 - p/2} \) in probability up to a factor (log log n)p/2. For \((F,G) \mapsto \smallint FdG\) and for \((F,G) \mapsto F \circ G^{ - 1} \) this yields nearly optimal remainder bounds. Also, p-variation gives new proofs for the asymptotic distributions of the Cramér-von Mises-Rosenblatt and Watson two-sample statistics when the two sample sizes m, n go to infinity arbitrarily.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUS

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