Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation
We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, 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 $$O\left(p^3\|A\|_4^2+pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2\right),$$ where $A\in\mathbb{R}^n$ collects the 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 add...
| Authors | Yixuan Zhang, Qiaomin Xie |
|---|---|
| Primary category | math.PR |
| Published | 8 September 2026 |
| Last updated at source | 11 October 2026 |
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