{
  "abstract": "Let $(F_n)$ be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that $F_n$ converges in law to a standard Gaussian vector. We prove that, for every integer $m\\geq0$ and every $p\\in[1,\\infty]$, the optimal rate of convergence of the densities in the Sobolev space $W^{m,p}$ is given by the maximum of the absolute third-order cumulants and the diagonal fourth-order cumulants. The same quantity also gives the optimal rates in total variation, Kolmogorov and $1$-Wasserstein distances. Our proof first derives, by Gaussian interpolation and Gaussian convolution, a cumulant expansion in the space of tempered distributions without imposing Malliavin nondegeneracy at the endpoint. Finite-order Malliavin density estimates then upgrade this identity to Sobolev spaces. Matching lower bounds follow from a finite-dimensional argument in which parity separates the third-order and fourth-order Gaussian corrections, while norm equivalence rules out cancellations among mixed cumulants. A superconvergence theorem supplies the required finite negative moments of the Malliavin determinant along a sufficiently far tail of the approximating sequence, so that no Malliavin nondegeneracy assumption is required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07066",
  "author_total": 3,
  "authors": [
    {
      "name": "Huiping Chen"
    },
    {
      "name": "Yong Chen"
    },
    {
      "name": "Yong Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9bejssanadp57.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07066",
  "fulltext_url": null,
  "id": "9bejssanadp57",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07066",
  "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",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7212060324",
  "pdf_url": "https://arxiv.org/pdf/2609.07066",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9bejssanadp57.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07066",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07066"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors",
  "title_text": "Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9bejssanadp57",
  "venue": "arXiv",
  "versions": []
}