{
  "abstract": "For multinomial counts $X\\sim\\operatorname{Mult}(N;\\mathbf p)$ on $n$ ordered bins and a reference law $\\mathbf r$ we study the expectation of the reference-centered discrete total variation $d_{\\mathbf r}(X)=\\sum_{j=1}^{n-1}|(X_{j+1}-Nr_{j+1})-(X_j-Nr_j)|$, the natural extension of the comb-test statistic to a non-uniform null. For a non-uniform null the raw total variation of the counts carries a deterministic staircase of order $N$ that no centering of the statistic removes; the counts are therefore centered at the reference law, and under the null ($\\mathbf p=\\mathbf r$) the statistic is the matched $d_{\\mathbf p}$. Exact finite-sample expectations are obtained by conditioning on the total mass in each adjacent pair, which reduces every edge to a translated binomial absolute-deviation problem. For fixed $n\\ge3$ and a sampling law bounded away from the boundary of the simplex, the matched statistic admits a three-term asymptotic expansion in powers of $1/N$ whose coefficients are explicit continuous periodic functions of $N(p_{j+1}-p_j)$ (a generalized, non-Poincaré asymptotic expansion), obtained by a weighted local Edgeworth estimate and a corner Euler--Maclaurin summation; all half-integer orders vanish by parity. Each correction separates into a smooth Edgeworth part and an oscillatory lattice part, the latter an Edgeworth perturbation of one formal Euler--Maclaurin series of a Gaussian lattice sum. Under a sampling law $\\mathbf p\\neq\\mathbf r$ the leading expectation is the folded-normal profile implied by the central limit theorem; under local alternatives we identify the $O(1)$ correction (a skewness term, plus a variance-gradient term for Pitman alternatives), and fixed alternatives give a law-of-large-numbers limit, the $\\ell_1$ distance between the adjacent-gradient profiles of $\\mathbf p$ and $\\mathbf r$, on which threshold tests are consistent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41255",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikola Banić"
    },
    {
      "name": "Neven Elezović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/955m7bdxqts1c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41255",
  "fulltext_url": null,
  "id": "955m7bdxqts1c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41255",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41255",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/955m7bdxqts1c.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41255",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41255"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Discrete total variation under the multinomial, Part 3: expectation under a general distribution",
  "title_text": "Discrete total variation under the multinomial, Part 3: expectation under a general distribution",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/955m7bdxqts1c",
  "venue": "arXiv",
  "versions": []
}