Frechet Differentiability, $p$-Variation and Uniform Donsker Classes

Abstract

Differentiability of functionals of the empirical distribution function is extended. The supremum norm is replaced by $p$-variation seminorms, which are the $p$th roots of suprema of sums of $p$th powers of absolute increments of a function over nonoverlapping intervals. Frechet derivatives often exist for such norms when they do not for the supremum norm. For $1 < q < 2$, classes of functions uniformly bounded in $q$-variation are universal and uniform Donsker classes: The central limit theorem for empirical measures holds with respect to uniform convergence over such a class, also uniformly over all probability laws on the line. The integral $\int F dG$ was defined by L. C. Young if $F$ and $G$ are of bounded $p$- and $q$-variation respectively, where $p^{-1} + q^{-1} > 1$. Thus the normalized empirical distribution function $n^{1/2}(F_n - F)$ is with high probability in sets of uniformly bounded $p$-variation for any $p > 2$, uniformly in $n$.