The p-variation of partial sum processes and the empirical process

• 成果类型: Article
• 署名作者: Qian, J
• 署名单位: Bucknell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 1370-1383
• 关键词: order

摘要

The p-variation of a function f is the supremum of the sums of the pth powers of absolute increments of f over nonoverlapping intervals. Let F be a continuous probability distribution function. Dudley has shown that the p-variation of the empirical process is bounded in probability as n --> infinity if and only if p > 2, and for 1 less than or equal to p less than or equal to 2, the p-variation of the empirical process is at least n(1-p/2) and is at most of the order n(1-p/2)(log log n)(p/2) in probability. In this paper, we prove that the exact order of the X-variation of the empirical process is log log n in probability, and for 1 less than or equal to p < 2, the p-variation of the empirical process is of exact order n(1-p/2) in expectation and almost surely. Let S-j := X1 + X-2 + ... + X-j. Then the p-variation of the partial sum process for {X-1, X-2, ..., X-n} is defined as that of f on (0, n], where f(t) = S-j for j - 1 < t less than or equal to j, j = 1, 2,..., n. Bretagnolle has shown that the expectation of the p-variation for independent centered random variables X-i with bounded pth moments is of order n for 1 p < 2. We prove that for p = 2, the 2-variation of the partial sum process of i.i.d. centered nonconstant random variables with finite 2 + delta moment for some delta > 0 is of exact order n log log n in probability.