{
  "abstract": "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 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09480",
  "author_total": 2,
  "authors": [
    {
      "name": "Yixuan Zhang"
    },
    {
      "name": "Qiaomin Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7j9y2t1f71nr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09480",
  "fulltext_url": null,
  "id": "7j9y2t1f71nr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09480",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7212171236",
  "pdf_url": "https://arxiv.org/pdf/2609.09480",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7j9y2t1f71nr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.09480",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09480"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation",
  "title_text": "Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/7j9y2t1f71nr9",
  "venue": "arXiv",
  "versions": []
}