Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains

arXiv:2609.09480v1 Announce Type: cross Abstract: We develop Gaussian approximation bounds in higher-order Wasserstein distance W_p, pgeq2, 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 $ Oleft( p^3 |A|_4^2 + pd^{1/4}|A|_2^{1/2}|A|_4^2 right) where AinR^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.

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Source: arXiv cs.LG | 2026-09-10