{
  "schema_version": 1,
  "problem_number": "OWR-13677-003",
  "title": "Purely Singular Copulas Are Typical for the Markov-Kernel Metric D1",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Durante, Fernández-Sánchez and Trutschnig (Oberwolfach Report 20/2015) asked whether the class of purely singular d-dimensional copulas is of second category in (C_d, D_1), where D_1 is the metric of Trutschnig defined through the conditional distributions (Markov kernels) of the copulas. The answer is yes: for every d ≥ 2 the purely singular copulas form a dense G_δ set in (C_d, D_1), hence a co-meager set. This follows in a few lines from known results, and we claim no priority for it: the purely singular copulas form a G_δ set already for the coarser uniform metric, (C_d, D_1) is complete, and the check-min approximation B_N(A) of a copula A, which carries the mass of each cell of the N-grid uniformly on the main diagonal of the cell, is purely singular and tends to A in D_1. For d = 2 this density is classical; Durante, Fernández-Sánchez and Trutschnig use it in \"A typical copula is singular\" (2015), citing Mikusiński and Taylor (Metrika 2010), and with Theorem 4.1 of that paper it settles the case d = 2, which the paper nevertheless lists as open. For d ≥ 3 the density may also be in Mikusiński and Taylor (2010), a paper on approximations of n-copulas that we could not read; we prove it with the bound D_1(A, B_N(A)) ≤ ‖H_A − P_N H_A‖_1 + (d − 1)/N, where H_A is the conditional distribution function and P_N averages over the columns of the grid. New, to our knowledge, is that the copulas whose conditional distributions are, for almost every value of the conditioning coordinate, singular and without atoms also form a dense G_δ set in (C_d, D_1): in this metric a typical copula has diffuse singular conditional distributions, whereas for d = 2 and the uniform metric they are typically point masses. If D_1 is replaced by the L^1 distance of the copula functions, as in the restatement of the question in version 1.7.0 of the UnsolvedMath collection, the question reduces to the known result for the uniform metric. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.ST",
    "math.FA"
  ],
  "keywords": [
    "copulas",
    "purely singular copulas",
    "Baire category",
    "co-meager sets",
    "Markov kernels",
    "metric D1",
    "doubly stochastic measures",
    "doubly stochastic operators",
    "check-min approximations",
    "checkerboard copulas",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-13677-003",
    "math.PR",
    "math.ST",
    "math.FA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13677-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-13677-003/paper.pdf?v=069ed6a8caea",
  "doi": "10.5281/zenodo.23372815",
  "zenodo_record_url": "https://zenodo.org/records/23372815",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the question of F. Durante, J. Fernández-Sánchez and W. Trutschnig (Oberwolfach Report 20/2015): for every d ≥ 2 the purely singular copulas form a dense G_δ set in (C_d, D_1). This follows in a few lines from known results and no priority is claimed for it; the density step for d ≥ 3 may be in Mikusiński and Taylor (2010), which could not be read. New, to our knowledge: copulas with singular atomless conditional distributions are also typical in D_1. Novelty is not certified.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}