Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains

Abstract

We develop Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\ge 2$, for sums of multivariate martingale differences generated by a uniformly ergodic Markov chain. Under an $L^{(2+\eta )p}$-moment condition with $\eta >0$, we establish the explicit bound $$O\left(p^3\Vert A{\Vert }_4^2+p{d}^{1/4}\Vert A{\Vert }_2^{1/2}\Vert A{\Vert }_4^2\right)$$ where $A\in {\mathbb{R}}^n$ collects the $L^{(2+\eta )p}$-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.

28 pages