  We provide explicit nonasymptotic estimates for the rate of convergence of
empirical means of Markov chains, together with a Gaussian or exponential
control on the deviations of empirical means. These estimates hold under a
"positive curvature" assumption expressing a kind of metric ergodicity, which
generalizes the Ricci curvature from differential geometry and, on finite
graphs, amounts to contraction under path coupling.
