Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation
Abstract
We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance Wp, p≥2, for deterministic matrix-weighted sums of multivariate Poisson martingale differences generated by uniformly ergodic Markov chains. Under an L(2+η)p-moment condition, η>0, we obtain the explicit bound O(p3|A|42+pd1/4|A|21/2|A|42), where A∈ℝn collects the moment sizes of the weighted increments. When these sizes are of order n−1/2, the bound achieves the first optimal O(n−1/2) rate for fixed p and d. Consequently, we also obtain the first optimal O(n−1/2) Wp 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. We apply these bounds to nonlinear Markovian stochastic approximation by decomposing normalized errors into matrix-weighted Poisson martingale sums and remainders. This yields the first sharp Wp Gaussian approximation rates for both last iterates and Polyak--Ruppert averages. In particular, the optimized rates are O(n−1/3) for polynomial-stepsize last iterates and O(n−1/4) for Polyak--Ruppert averages, with an O(n−1/2) rate for harmonic-stepsize last iterates under an additional spectral condition.
Related
- Berry--Esseen bounds for stochastic approximation with Markovian noise
- Shrinking-Tube Concentration for Adaptive Markovian Stochastic Approximation
- Martingale central limit theorems in p-Wasserstein distance
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- Càdlàg Solutions to Backward Stochastic Dynamics featuring Oblique Subgradients and driven by Martingale Noise
BibTeX
@misc{zhang2026gaussian,
title = {{Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation}},
author = {Yixuan Zhang and Qiaomin Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.09480},
eprint = {2609.09480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09480},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation
AU - Yixuan Zhang
AU - Qiaomin Xie
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.09480
UR - https://arxiv.org/abs/2609.09480
AB - We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance W_p, p≥2, for deterministic matrix-weighted sums of multivariate Poisson martingale differences generated by uniformly ergodic Markov chains. Under an L^(2+η)p-moment condition, η>0, we obtain the explicit bound O(p^3|A|_4^2+pd^1/4|A|_2^1/2|A|_4^2), where A∈ℝ^n collects the moment sizes of the weighted increments. When these sizes are of order n^−1/2, the bound achieves the first optimal O(n^−1/2) 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. We apply these bounds to nonlinear Markovian stochastic approximation by decomposing normalized errors into matrix-weighted Poisson martingale sums and remainders. This yields the first sharp W_p Gaussian approximation rates for both last iterates and Polyak--Ruppert averages. In particular, the optimized rates are O(n^−1/3) for polynomial-stepsize last iterates and O(n^−1/4) for Polyak--Ruppert averages, with an O(n^−1/2) rate for harmonic-stepsize last iterates under an additional spectral condition.
ER -