Central limit theorem for Markov processes with spectral gap in the Wasserstein metric

Abstract

Suppose that $\{X_t,\,t\ge0\}$ is a non-stationary Markov process, taking values in a Polish metric space $E$. We prove the law of large numbers and central limit theorem for an additive functional of the form $\int_0^T\psi(X_s)ds$, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function $\psi$ is assumed to be Lipschitz on $E$.