  In this paper, we derive asymptotic results for L^1-Wasserstein distance
between the distribution function and the corresponding empirical distribution
function of a stationary sequence. Next, we give some applications to dynamical
systems and causal linear processes. To prove our main result, we give a
Central Limit Theorem for ergodic stationary sequences of random variables with
values in L^1. The conditions obtained are expressed in terms of
projective-type conditions. The main tools are martingale approximations.
