Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation

We develop Gaussian approximation bounds in higher-order Wasserstein distance W p, p at least2, for sums of multivariate martingale differences generated by a uniformly ergodic Markov chain. Under an L(2+η)p-moment condition with η>0, we establish the explicit bound O (p cubed A 4 squared + pd 1/4A 2 1/2A 4 squared) where A in R n collects the L(2+η)p-sizes of the n individual martingale increments. In the balanced-increment regime where the individual increments have comparable sizes of order n-1/2, it yields the first optimal O(n-1/2) Gaussian approximation rate for fixed p and d. Consequently, we also obtain the first optimal O(n-1/2) W p Gaussian approximation rate for multivariate additive functionals of uniformly ergodic Markov chains. Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence. First, building on the Ornstein--Uhlenbeck relative-score approach of Fang and Koike (2023), we formulate the bound in terms of antisymmetric Stein couplings while retaining the conditional tensor structure. Second, we develop a refresh-then-maximal coupling that combines an independent first-step resampling, which preserves the desired Stein identity, with a subsequent maximal coupling that provides effective control of the coupling increment. These tools may be useful more broadly for Gaussian approximation under temporal dependence.