{
  "abstract": "We prove that for the normalized sum $W=\\sum_{i=1}^n X_i/\\sqrt{n}$ of independent, mean-zero, $d$-dimensional random vectors with $\\operatorname{Cov}(W) =Σ=\\operatorname{diag}(σ_1^2,\\dots, σ_d^2)$ (the diagonal form of the covariance matrix is assumed without loss of generality), the 2-Wasserstein distance from $W$ to $N(0,Σ)$ is bounded by \\[\\sqrt{2\\sum_{j=1}^d \\min \\left\\{σ_j^2, \\frac{\\sum_{i=1}^n \\mathbb{E} [|X_i'-X_i|^2(X'_{ij}-X_{ij})^2 ] }{n^2 σ_j^2} \\right\\}},\\] where $|\\cdot|$ denotes the Euclidean norm, $X_i'$ is an independent copy of $X_i$, and $X_{ij}$ ($X_{ij}'$ resp.) denotes the $j$th coordinate of the vector $X_i$ ($X_i'$ resp.). On the one hand, it gives the optimal rate $O(β\\sqrt{d/n})$ if $|X_i|\\leq β$ for all $i$. On the other hand, in the isotropic case where $Σ=I_d$, it gives the optimal fourth-moment bound $O\\left(\\sqrt{\\sum_{i=1}^n \\mathbb{E}|X_i|^4/n^2}\\right)$. Moreover, the coordinatewise minima suggest a tradeoff between the naïve coupling and the normal coupling for each component. To prove the main result, we develop a new transportation inequality using a coupling via the Föllmer process with a general covariance matrix and the martingale property of its drift. The resulting bound is then controlled using a Bonis-type expansion together with Bessel's inequality. The method and the result extend to $p$-Wasserstein bounds. As a consequence, we obtain the Hilbert space 4-Wasserstein CLT bound $O(n^{-1/4}[\\operatorname{tr}(Q^{1/2})]^{1/2})$, where, in the i.i.d. case, $Q=\\mathbb{E}[|X_1'-X_1|^2(X_1'-X_1)^{\\otimes2}]$ and the $O(n^{-1/4})$ rate is optimal given the dimension-free form of $Q$. We further strengthen it to a dimension-free strong approximation result with a near-optimal error rate through the KMT embedding construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35354",
  "author_total": 4,
  "authors": [
    {
      "name": "Xiao Fang"
    },
    {
      "name": "Zi-Yao Su"
    },
    {
      "name": "Jiaxue Wang"
    },
    {
      "name": "Xiaolin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ddwaxz6wwjda0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35354",
  "fulltext_url": null,
  "id": "ddwaxz6wwjda0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35354",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.35354",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ddwaxz6wwjda0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35354",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35354"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Fourth-Moment Wasserstein CLT Bounds",
  "title_text": "Sharp Fourth-Moment Wasserstein CLT Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/ddwaxz6wwjda0",
  "venue": "arXiv",
  "versions": []
}