{
  "abstract": "Let $X\\sim Bin(N,p)$ and let $c=Nr$ be a prescribed centre with $r\\ne p$. The finite formula for $E|X-c|$ is classical, and the existence of a complete large-deviation expansion in powers of $N^{-1}$ follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in closed form for the stop-loss tail produced by the absolute deviation, with the lattice defect of $Nr$ retained explicitly. The construction combines three standard ingredients: Bernoulli polynomials for the logarithm of successive binomial masses, the exponential Bell recursion, and Eulerian polynomials for weighted geometric sums. The $k$-th coefficient has denominator exactly $(1-ρ)^{2k+2}$, where $ρ$ is the exponential tilt; hence the effective expansion parameter is $1/(N(1-ρ)^2)$, the lattice weighted analogue of the parameter in the incomplete gamma expansion. The case $r=\\tfrac12$ gives the fold at the origin of a biased $\\pm1$ random walk, and its parity split is identified with the lattice defect of the fold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19064",
  "author_total": 1,
  "authors": [
    {
      "name": "Neven Elezović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9pnkfv2zsma8m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19064",
  "fulltext_url": null,
  "id": "9pnkfv2zsma8m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19064",
  "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.CA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7213516865",
  "pdf_url": "https://arxiv.org/pdf/2609.19064",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9pnkfv2zsma8m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19064",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19064"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form",
  "title_text": "Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/9pnkfv2zsma8m",
  "venue": "arXiv",
  "versions": []
}