Central Limit Theorems for Markov Chains from Wasserstein Convergence Rates

Abstract

We give sufficient conditions for central limit theorems (CLTs) for additive functionals of Markov chains in terms of quantitative Wasserstein convergence rates, including suitable subgeometric rates. For a given metric $\psi$, we establish CLTs for $\psi$-Lipschitz functions under moment conditions by showing that suitable $1$-Wasserstein convergence rates imply either the Maxwell--Woodroofe projective criterion or convergence of the associated Poisson series. We then extend this framework beyond the $\psi$-Lipschitz setting in two directions. First, by reweighting $\psi$ with a non-negative function $V$, we construct a weighted path metric under which functions with $V$-controlled increments are Lipschitz. We derive convergence bounds in the Wasserstein distance induced by this new metric from corresponding bounds in the Wasserstein distance induced by $\psi$, thereby obtaining CLTs for this broad class of functions. Second, we consider functions admitting integrable increment envelopes and derive CLTs from quantitative $2$-Wasserstein convergence rates. On $\mathbb R^d$, pointwise Sobolev inequalities and polynomial bounds on the gradient provide concrete sufficient conditions for constructing such envelopes. We illustrate the results with nonlinear autoregressive processes and a random walk on the one-dimensional torus exhibiting subgeometric Wasserstein convergence.