{
  "abstract": "We study the expectation of the discrete total variation statistic $d=\\sum_{i=1}^{n-1}|X_{i+1}-X_i|$ of counts $X_1,\\dots,X_n$ over $n$ ordered bins with sample size $N$, under the uniform multinomial null. In the notation used throughout the series, a reference distribution $\\mathbf r$ defines $d_{\\mathbf r}(x)=\\|x-N\\mathbf r\\|_V$; here $\\mathbf r=\\mathbf u$ is uniform and we write simply $d$. We obtain an exact single-sum formula, a closed form in central trinomial coefficients valid for every $n\\ge2$, with one-term evaluations at $n=2,4$ and a complete D-finite characterization for $n=3$. The per-edge expectation $W_N=E|X_2-X_1|$ satisfies an exact three-term recurrence, from which we derive its complete $1/N$ expansion with the $n$-independent constant $E(d^\\circ/\\sqrt{Nn})\\to 2/\\sqrtπ$ for the cyclic statistic $d^\\circ=d^\\circ_{\\mathbf u}$, independent of the number of bins; the complementary sparse regime (fixed $N$, $n\\to\\infty$) is treated in a separate paper. Conditioning on a two-bin total reduces each edge to a symmetric random walk; as the subsequent parts of the series will show, this method carries the analysis to non-uniform reference laws.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13366",
  "author_total": 2,
  "authors": [
    {
      "name": "N. Banić"
    },
    {
      "name": "N. Elezović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8cq1hk7d4q3cz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13366",
  "fulltext_url": null,
  "id": "8cq1hk7d4q3cz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13366",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7221555912",
  "pdf_url": "https://arxiv.org/pdf/2610.13366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8cq1hk7d4q3cz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Discrete total variation under the multinomial, Part~1: expectation under the uniform null",
  "title_text": "Discrete total variation under the multinomial, Part~1: expectation under the uniform null",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/8cq1hk7d4q3cz",
  "venue": "arXiv",
  "versions": []
}