Gaussian approximation improves understanding of dependent data sums
Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains
Machine Learning
Summary
This paper studies how to approximate the sum of complex dependent data points, specifically those coming from a type of random process called a uniformly ergodic Markov chain. The authors found a way to measure how close this sum is to a multivariate normal (Gaussian) distribution in a strong mathematical sense. Their work establishes the best possible rate at which this approximation improves as more data points are summed, under certain technical conditions. They also developed novel mathematical techniques to handle dependencies over time and to keep the approximation accurate.
What this means in practice
- •For machine learning engineers: Enhance confidence estimates in algorithms that rely on dependent temporal data by providing tighter error bounds for distribution approximations.
- •For financial risk analysts: Improve modeling of dependent risk factors by accurately approximating sums of time-dependent stochastic processes with Gaussian distributions.
Authors
Yixuan Zhang, Qiaomin Xie
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.