Convergence Rates for Empirical Measures of Markov Chains in Dual and Wasserstein Distances

We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $\mu$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular the $1$-Wasserstein distance. The main result of this article is a new upper bound for the expected distance, which is proved by combining a Fourier expansion with a truncation argument. Our bound matches the known rates for i.i.d. random variables up to logarithmic factors. In addition, we show how concentration inequalities around the mean can be obtained.