Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains

2026-09-08

AI Summary

本文解决了由均匀遍历马尔可夫链生成的多变量鞅差和的高斯逼近问题，通过在Wasserstein距离下建立显式界来解决。

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.

Problem

• Gaussian Approximation
• Multivariate Martingale Sums
• Uniformly Ergodic Markov Chains
• Wasserstein Distance

Innovation

• Gaussian Approximation
• Higher-order Wasserstein Distance
• Uniformly Ergodic Markov Chains
• Antisymmetric Stein Couplings
• Refresh-then-maximal Coupling