FRECHET DIFFERENTIABILITY, P-VARIATION AND UNIFORM DONSKER CLASSES

• 成果类型: Article
• 署名作者: DUDLEY, RM
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989537
• 发表日期: 1992
• 页码: 1968-1982
• 关键词: theorem

摘要

Differentiability of functionals of the empirical distribution function is extended. The supremum norm is replaced by p-variation seminorms, which are the pth roots of suprema of sums of pth 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 integral 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 n1/2(F(n) - F) is with high probability in sets of uniformly bounded p-variation for any p > 2, uniformly in n.