{
  "abstract": "Bergsma--Dassios sign covariance $τ^*$ is a rank-based population measure of dependence. Building on zero-characterisation results under specific regularity regimes, we prove that $τ^*(X,Y)=0$ characterises independence for every real-valued bivariate distribution, including mixed and singular laws. For the unnormalised four-sample convention for $τ^*$ defined in Subsection 4.3 and the unscaled Blum--Kiefer--Rosenblatt functional $\\mathscr {B}$, the proof gives the quantitative inequality $τ^*\\ge 2\\mathscr {B}$. This is a population identification result; no new sample-level limit theorem is claimed. The argument first encodes finite ordered distributions with rational cell probabilities by labelled path trees and applies a nonnegative sum-of-squares representation for a quartet covariance. Rational approximation and nested quantisation then remove all support and regularity restrictions. On finite uniformly weighted label sets, the tree framework also relates an edge-weighted quartet quantity to empirical distance covariance squared. As a separate combinatorial consequence, it yields the asymptotic $2/3$ upper bound for the quartet distance between binary phylogenetic trees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30331",
  "author_total": 2,
  "authors": [
    {
      "name": "Stefan Grünewald"
    },
    {
      "name": "Libo Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e50xxex71jjp7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30331",
  "fulltext_url": null,
  "id": "e50xxex71jjp7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30331",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7204887899",
  "pdf_url": "https://arxiv.org/pdf/2608.30331",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e50xxex71jjp7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30331",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30331"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws",
  "title_text": "Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:54Z",
  "url": "https://research.draconitic.com/paper/e50xxex71jjp7",
  "venue": "arXiv",
  "versions": []
}