{
  "abstract": "Let $(S_n)_{n\\geq 1}$ be the one-dimensional Elephant Random Walk (ERW) in the superdiffusive regime, with memory parameter $p\\in(3/4,1)$ and first-step bias $\\mathbb{P}(S_1=1)=q$. Set $a=2p-1\\in(1/2,1)$. If $f_{q,a}$ denotes the density of the superdiffusive limit $L_{q,a}$, we prove the uniform local limit theorem $\\lim_{n\\to\\infty}\\sup_{j\\in\\mathbb{Z},\\,j\\equiv n\\;(\\mathrm{mod}\\;2)}\\left|\\frac{n^a}{2}\\mathbb{P}(S_n=j)-f_{q,a}(j/n^a)\\right|=0$. The proof uses the exact recurrence for the probability mass function of $S_n$, yielding uniform $O(n^{-a})$ and $O(n^{-2a})$ bounds for the p.m.f. and its first discrete differences, respectively. We then disprove a conjecture of Guérin, Laulin, Raschel and Simon concerning eventual log-concavity. Writing $L_a=L_{1,a}$ and $f_a=f_{1,a}$, let $a_{\\rm ev}$ be the eventual-log-concavity threshold and let $a_\\star$ be the upper failure threshold for log-concavity of $f_a$. Their results imply $a_{\\rm ev}\\geq(\\sqrt{5}-1)/2$, and we prove $(\\sqrt{5}-1)/2\\leq a_{\\rm ev}\\leq\\min\\{0.80399,a_\\star\\}\\leq a_\\star<0.918$. The bound $a_\\star<0.918$ follows from the first three exact moments of $L_a$ and the sharp skewness inequality for centered log-concave distributions, while $a_{\\rm ev}\\leq0.80399$ follows from a certified finite-order analysis of the exact p.m.f. recurrence. Direct numerical iterations provide evidence that $a_{\\rm ev}\\approx0.80399$, equivalently $p_{\\rm ev}\\approx0.902$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14753",
  "author_total": 2,
  "authors": [
    {
      "name": "Hélio Trinas"
    },
    {
      "name": "Glauco Valle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ccgtrv3d7a148.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14753",
  "fulltext_url": null,
  "id": "ccgtrv3d7a148",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14753",
  "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.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213340339",
  "pdf_url": "https://arxiv.org/pdf/2609.14753",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ccgtrv3d7a148.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14753",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14753"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity",
  "title_text": "A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/ccgtrv3d7a148",
  "venue": "arXiv",
  "versions": []
}