# 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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