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

Suppose that { X t , t ≥ 0 } 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 ∫ 0 T ψ ( X s ) d s , provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function ψ is assumed to be Lipschitz on E .