Strong laws for local quantile processes

• 成果类型: Article
• 署名作者: Deheuvels, P
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1023481119
• 发表日期: 1997
• 页码: 2007-2054
• 关键词: strong limit-theorems; iterated logarithm; functional laws; Empirical Process; partial sums; approximation

摘要

We show that increments of size h(n) from the uniform quantile and uniform empirical processes in the neighborhood of a fixed point t(0) is an element of (0, 1) may have different rates of almost sure convergence to 0 in the range where h(n) --> 0 and nh(n)/log n --> infinity. In particular, when h(n) + n(-lambda) with 0 < lambda < 1, we obtain that these rates are identical for 1/2 < lambda < 1, and distinct for 0 < lambda < 1/2. This phenomenon is shown to be a consequence of functional laws of the iterated logarithm for local quantile processes, which we describe in a more general setting. As a consequence of these results, we prove that, for any epsilon > 0, the best possible uniform almost sure rate of approximation of the uniform quantile process by a normed Kiefer process is not better than O(n(-1/4)(log n)(-epsilon)).