Wasserstein Rate Driven CLTs for Markov Chains with Weighted Lipschitz, Sobolev, and Stein Test Functions

Abstract

Quantitative convergence in Wasserstein distance is often easier to establish than that in total variation distance. We show that such bounds allowing subgeometric rates yield central limit theorems (CLTs) for additive functionals of Markov chains without converting to total variation distance. Specifically, for a metric $\psi$, we derive two CLTs for $\psi$-Lipschitz observables under mild moment assumptions by verifying the Maxwell-Woodroofe and Poisson-series criteria directly from Wasserstein rates. We then enlarge the admissible classes via two lifts: (i) a weighted path-metric construction giving CLTs for weighted-Lipschitz functions with controlled polynomial growth; (ii) an analytic $W_2$ route yielding $L^2(\pi)$ decay of the $k$ step expectation bias, which in turn gives CLTs for a weighted Sobolev class and for Stein test functions, together with a computable comparison between the kernelized Stein discrepancy (KSD) and $W_2$ rates, namely $\mathrm{KSD}\lesssim W_2$. The framework accommodates subgeometric mixing and certain reducible chains. Examples include nonlinear autoregressive processes, an Ornstein-Uhlenbeck chain, and a reducible AR(1) model.