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−δ for some positive δ. This shows that
our conditions are in some sense optimal.Comment: Published in at http://dx.doi.org/10.1214/12-AOP798 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org