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

TL;DR

This work addresses central limit theorems for additive functionals of Markov chains using Wasserstein convergence rates, including subgeometric rates, thereby avoiding total variation. It introduces a two-stage pipeline: Stage I converts quantitative Wasserstein bounds into CLTs for $\psi$-Lipschitz observables via Maxwell–Woodroofe and Poisson-series criteria, and Stage II provides principled lifts to broader observables through a weighted path-metric (weighted Lipschitz) and an analytic $W_2$ (Sobolev/Stein) framework, with a computable $KSD \lesssim W_2$ relation. Key contributions include new martingale-approximation results from Wasserstein rates, CLTs for subgeometric and reducible chains, and extended CLTs for weighted Lipschitz, Sobolev, and Stein test functions, plus connections to Stein diagnostics. The approach is demonstrated on nonlinear autoregressive processes, an Ornstein–Uhlenbeck chain, and reducible AR(1) models, yielding practical tools for high-dimensional MCMC diagnostics and variance reduction through KSD-based methods while broadening the class of observables beyond uniformly Lipschitz 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 $ψ$, we derive two CLTs for $ψ$-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 $L^2(π)$ 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.

Theorems & Definitions (67)

• Definition 1
• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3
• Lemma 2
• Proposition 1
• Proposition 2
• Remark 1
• Theorem 4