{
  "abstract": "We construct a symmetric operad $\\mathfrak{C}$ on the class of all multivariate copulas, where operadic composition is given by Sklar substitution. We prove that the absolutely continuous subclass $\\mathfrak{C}^{ac}$---which coincides with the $L^1$ class of copula densities---forms a suboperad; under composition, the density of the composite copula is given by the explicit Sklar substitution density formula $g(v)=ϕ\\big(Ψ_1(v^{(1)}),\\dots,Ψ_n(v^{(n)})\\big)\\prod_{k=1}^{n}ψ_{k}(v^{(k)})$. Furthermore, we show that copulas with finite copula entropy---identified with the $L\\log L$ class of copula densities---are closed under substitution and hence constitute a suboperad $\\mathfrak{C}^{L\\log L}$. On this suboperad, copula entropy is strictly additive: $H(γ(Φ;Ψ_{1},\\ldots,Ψ_{n}))=H(Φ)+\\sum_{k=1}^{n}H(Ψ_{k})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20512",
  "author_total": 1,
  "authors": [
    {
      "name": "Xuexing Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eewx37s20ncqh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20512",
  "fulltext_url": null,
  "id": "eewx37s20ncqh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20512",
  "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.IT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
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      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
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      "arxiv",
      "stat.TH"
    ],
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      "openalex",
      "2003|Financial Risk and Volatility Modeling"
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  "openalex": "W7213591201",
  "pdf_url": "https://arxiv.org/pdf/2609.20512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eewx37s20ncqh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.20512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Copula Operad and Copula Entropy",
  "title_text": "Copula Operad and Copula Entropy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/eewx37s20ncqh",
  "venue": "arXiv",
  "versions": []
}