An empirical Central Limit Theorem in L1 for stationary sequences

In this paper, we derive asymptotic results for the L1-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 L1. The conditions obtained are expressed in terms of projective-type conditions. The main tools are martingale approximations.