Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains

• URL: http://arxiv.org/abs/2609.09480v1
• Date: Tue, 08 Sep 2026 21:54:29 GMT
• Title: Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains
• Abstract summary: We develop bounds in higher-order Wasserstein distance $W_p$, $pgeq2$, for sums of multivariate martingale differences generated by a uniformly ergodic Markov chain.<n>Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence.
• Score: 19.40811084751516
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We develop Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, 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 $$ O\left( p^3 \|A\|_4^2 + pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2 \right) $$ where $A\in\mathbb{R}^n$ 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.

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