  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.
