  The main purpose of this paper is to investigate the strong approximation of
the $p$-fold integrated empirical process, $p$ being a fixed positive integer.
More precisely, we obtain the exact rate of the approximations by a sequence of
weighted Brownian bridges and a weighted Kiefer process. Our arguments are
based in part on results of Koml\'os, Major and Tusn\'ady (1975). We also
obtain an exponential bound for the tail probability of the weighted
approximation to the $p$-fold integrated empirical process. Applications
include the two-sample testing procedures together with the change-point
problems. We also consider the strong approximation of integrated empirical
processes when the parameters are estimated. We study the behavior of the
self-intersection local time of the partial sum process representation of
integrated empirical processes. Finally, simulation results are provided to
illustrate the finite sample performance of the proposed statistical tests
based on the integrated empirical processes.
