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

Miklós Csörgő, Barbara Szyszkowicz, Lihong Wang

The Annals of Statistics · 2006 · Volume 34 · Issue 2

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.

Keywords

60F15, 60F17, 60G10, 60G18, long-range dependence, sequential Bahadur–Kiefer process, sequential empirical and quantile processes, sequential Vervaat and Vervaat error processes, Strong invariance principles

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