  We study the problem of non-asymptotic deviations between a reference measure
and its empirical version, in the 1-Wasserstein metric, under the standing
assumption that the measure satisfies a transport-entropy inequality. We extend
some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our
methods are based on concentration inequalities and extend to the general
setting of measures on a Polish space. Deviation bounds for the occupation
measure of a Markov chain are also given, under the assumption that the chain
is contractive on the space of Lipschitz functions. Throughout the text,
several examples are worked out, including the cases of Gaussian measures on
separable Banach spaces, and laws of diffusion processes.
