Speed of convergence of classical empirical processes in p-variation norm
成果类型：
Article
署名作者：
Huang, YC; Dudley, RM
署名单位：
Massachusetts Institute of Technology (MIT)
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
2001
页码：
1625-1636
关键词：
摘要：
Let F be any distribution function on R, and F-n be the nth empirical distribution function based on variables i.i.d. (F). It is shown that for 2 < p < infinity and a constant C(p) < infinity, not depending on F, on some probability space there exist Fn and Brownian bridges B-n such that for the Wiener-Young p-variation norm parallel to . parallel to([p]), Eparallel ton(1/2)(F-n - F) - B-n o Fparallel to([P]) less than or equal to C(p)n((2-p)/(2p)), where (B-n o F)(x) = B-n(F(x)). The expectation can be replaced by an Orliez norm of exponential order. Conversely, if F is continuous, then for any stochastic process V(t, omega) continuous in t for almost all w, such as Bn o F, summation over n distinct jumps shows that parallel ton(1/2)(F-n - F) - Vparallel to([p]) greater than or equal to n((2-p)1(2P)), so the upper bound in expectation is best possible up to the constant C(p). In the proof, B-n is linked to F-n by the Komlos, Major and Tusnady construction, as for the supremum norm (p = infinity).