{
  "abstract": "We construct tie-symmetrised extensions $\\widetilde D$ and $\\widetilde R$ of the unnormalised Hoeffding and Blum--Kiefer--Rosenblatt functionals $D$ and $R$ and prove, for every real-valued bivariate law, the exact nonnegative decomposition $ τ^{*}(X,Y)=12\\widetilde D(X,Y)+24\\widetilde R(X,Y), \\qquad \\widetilde D,\\widetilde R\\geq0, \\qquad \\widetilde R(X,Y)=0\\quad\\Longleftrightarrow\\quad X\\perp\\!\\!\\!\\perp Y. $ For atomless margins the components reduce to $D$ and $R$, whereas in the presence of ties the corresponding classical identity $τ^{*}=12D+24R$ can fail. The components arise from the Hoeffding projections of a symmetrised pair kernel. For the four strict/non-strict boundary versions $D_{ε,η}$ and $R_{ε,η}$, we establish the sharp comparisons $ 9\\widetilde D\\geq\\sum_{ε,η}D_{ε,η}, \\qquad 9\\widetilde R\\geq\\sum_{ε,η}R_{ε,η}. $ Consequently, $τ^{*}\\geq0$ for every bivariate law, and the Bergsma--Dassios conjecture is settled: $τ^{*}=0$ if and only if $X\\perp\\!\\!\\!\\perp Y$. The boundary comparisons also yield the universal bound $τ^{*}\\geq(8/3)R$. Finite-table arguments, including an exact sum-of-squares certificate, establish the boundary inequalities, and weak-order quantisation transfers them to arbitrary laws. The factor $9$ in each comparison and the coefficient $24$ in $τ^{*}\\geq24\\widetilde R$ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06529",
  "author_total": 2,
  "authors": [
    {
      "name": "Wicher Bergsma"
    },
    {
      "name": "Angelos Dassios"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ehy3kw0kt5yys.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06529",
  "fulltext_url": null,
  "id": "ehy3kw0kt5yys",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06529",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7212019221",
  "pdf_url": "https://arxiv.org/pdf/2609.06529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ehy3kw0kt5yys.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.06529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds",
  "title_text": "Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:36Z",
  "url": "https://research.draconitic.com/paper/ehy3kw0kt5yys",
  "venue": "arXiv",
  "versions": []
}