{
  "abstract": "Let $X\\sim\\text{Mult}(N;p_1,\\dots,p_n)$ with $p_i>0$ and $\\sum_ip_i=1$. We study $\\Pr\\{X=m\\}$ for lattice points near $Np$ from two complementary viewpoints. First, we give an exact integer-mode criterion: the multinomial mode is the Jefferson--D'Hondt divisor apportionment, given when the mode is unique by $m_i=\\lfloor (N+d)p_i\\rfloor$, $d\\in[0,n)$. For $n\\ge3$ the mode need not be obtained by rounding each $Np_i$ to a neighbouring integer; the excess $m_i-\\lceil Np_i\\rceil$ can reach $n-2$, so the two-sided localisation $|m_i-Np_i|<1$ stated in a standard reference is not valid in general. Second, by writing the mass as a gamma quotient with unequal scalings, we derive the complete local expansion of $\\Pr\\{X=m\\}$ in integer powers of $N^{-1}$, with all coefficients in closed Bernoulli-polynomial form. The expansion contains the known $O(N^{-1})$ local limit theorem, reduces to the binomial local mass for $n=2$ and to the central multinomial coefficient in the symmetric case, and admits Cesàro averages of the oscillating coefficients in closed $ζ$-value form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20229",
  "author_total": 1,
  "authors": [
    {
      "name": "Neven Elezović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/13635pxzjrzjg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20229",
  "fulltext_url": null,
  "id": "13635pxzjrzjg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20229",
  "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": "W7213593303",
  "pdf_url": "https://arxiv.org/pdf/2609.20229",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/13635pxzjrzjg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20229",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20229"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Multinomial probabilities near the mode: integer modes and the complete local expansion",
  "title_text": "Multinomial probabilities near the mode: integer modes and the complete local expansion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/13635pxzjrzjg",
  "venue": "arXiv",
  "versions": []
}