Strong invariance principles for sequential Bahadur-Kiefer and Vervaat error processes of long-range dependent sequences

• 成果类型: Article
• 署名作者: Csorgo, Miklos; Szyszkowicz, Barbara; Wang, Lihong
• 署名单位: Carleton University; Nanjing University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000164
• 发表日期: 2006
• 页码: 1013-1044
• 关键词: CENTRAL LIMIT-THEOREMS; iterated logarithm; Empirical Process; LAW; CONVERGENCE; quantiles

摘要

In this paper we study strong approximations (invariance principles) of the sequential uniform and general Bahadur-Kiefer processes of long-range dependent sequences. We also investigate the strong and weak asymptotic behavior of the sequential Vervaat process, that is, the integrated sequential Bahadur-Kiefer process, properly normalized, as well as that of its deviation from its limiting process, the so-called Vervaat error process. It is well known that the Bahadur-Kiefer and the Vervaat error processes cannot converge weakly in the i.i.d. case. In contrast to this, we conclude that the Bahadur-Kiefer and Vervaat error processes, as well as their sequential versions, do converge weakly to a Dehling-Taqqu type limit process for certain long-range dependent sequences.