The Annals of Statistics

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

Miklós Csörgő, Barbara Szyszkowicz, and Lihong Wang

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Abstract

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.

Article information

Source
Ann. Statist., Volume 34, Number 2 (2006), 1013-1044.

Dates
First available in Project Euclid: 27 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1151418250

Digital Object Identifier
doi:10.1214/009053606000000164

Mathematical Reviews number (MathSciNet)
MR2283402

Zentralblatt MATH identifier
1113.60034

Subjects
Primary: 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60G10: Stationary processes 60G18: Self-similar processes

Keywords
Long-range dependence sequential empirical and quantile processes sequential Bahadur–Kiefer process sequential Vervaat and Vervaat error processes strong invariance principles

Citation

Csörgő, Miklós; Szyszkowicz, Barbara; Wang, Lihong. Strong invariance principles for sequential Bahadur–Kiefer and Vervaat error processes of long-range dependent sequences. Ann. Statist. 34 (2006), no. 2, 1013--1044. doi:10.1214/009053606000000164. https://projecteuclid.org/euclid.aos/1151418250


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