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.
"Curvature, concentration and error estimates for Markov chain Monte Carlo." Ann. Probab. 38 (6) 2418 - 2442, November 2010. https://doi.org/10.1214/10-AOP541