{
  "abstract": "For the counts $(X_1,\\dots,X_n)\\sim\\mathrm{Mult}(N;1/n,\\dots,1/n)$ of $N$ observations in $n$ ordered bins we study the variance of the discrete total variation $d=\\sum_{i=1}^{n-1}|X_{i+1}-X_i|$. An exact decomposition splits $\\mathrm{Var}(d)$ into per-edge, adjacent-pair and disjoint-pair terms and yields, for fixed $n$ as $N\\to\\infty$, $\\mathrm{Var}(d)/N=α(n)-β(n)/N+γ(n)/N^2+o(N^{-2})$. For every $n\\ge 2$, $Nα(n)$ is the classical normal-theory variance (Hoel, Kamat) of the sum of absolute successive differences of $n$ independent $N(0,N/n)$ variables; the correction $β(n)$ is obtained in closed form for $n\\ge 3$ (the case $n=2$ is treated separately). Two disjoint edge differences $D_1=X_2-X_1$ and $D_3=X_4-X_3$ are exactly uncorrelated and asymptotically independent, yet $\\mathrm{Cov}(|D_1|,|D_3|)=-1/(πn)+O(1/N)$: a correlation of relative order $1/N$, invisible in the Gaussian limit and absent under Poissonized sampling, whose constant is proportional to the exact joint fourth cumulant $κ_{2,2}(D_1,D_3)=-4N/n^2$ and is proved here on the lattice. The adjacent pair contributes the lattice-Edgeworth constant $a_*=-2\\sqrt{3}/(9π)$, of which one quarter is a smooth Edgeworth term and three quarters a lattice-corner term; the disjoint skeleton $h(m)$ is closed through a Bessel generating function; and $γ(n)$ ($n\\ge 5$) is expressed through a single triangular-lattice constant whose closed form we conjecture. Together with the mean of Part 1, the variance supplies the two moments used by moment-matched approximations to the null law of $d$. This is Part 2 of a series; the cyclic statistic $d^{c}$ will be treated in Part 5.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17046",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikola Banić"
    },
    {
      "name": "Neven Elezović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3cpxgvtp056wj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17046",
  "fulltext_url": null,
  "id": "3cpxgvtp056wj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17046",
  "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"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7221779796",
  "pdf_url": "https://arxiv.org/pdf/2610.17046",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3cpxgvtp056wj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17046",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17046"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Discrete total variation under the multinomial, Part 2: variance under the uniform null",
  "title_text": "Discrete total variation under the multinomial, Part 2: variance under the uniform null",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/3cpxgvtp056wj",
  "venue": "arXiv",
  "versions": []
}