  We study the Wasserstein distance of order 1 between the empirical
distribution and the marginal distribution of stationary $\alpha$-dependent
sequences. We prove some moments inequalities of order p for any p $\ge$ 1, and
we give some conditions under which the central limit theorem holds. We apply
our results to unbounded functions of expanding maps of the interval with a
neutral fixed point at zero. The moment inequalities for the Wasserstein
distance are similar to the well known von Bahr-Esseen or Rosenthal bounds for
partial sums, and seem to be new even in the case of independent and
identically distributed random variables.
