Strong approximation results for the empirical process of stationary sequences

We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than $n^{-1-\delta}$ for some positive $\delta$. This shows that our conditions are in some sense optimal.