{
  "abstract": "Let $(S_n)$ be the one-dimensional elephant random walk with memory parameter $p\\in(3/4,1)$ and first-step probability $q=\\mathbb{P}(S_1=1)\\in[0,1]$. Set $a=2p-1$, and let $f_{q,a}$ denote the density of the almost-sure limit of $S_n/n^a$. We prove that the last return time $T_0=\\sup\\{m\\geq0:S_m=0\\}$ satisfies: $\\mathbb{P}(T_0\\geq n)\\sim f_{q,a}(0)\\sqrt{\\frac{8}{π(2a-1)}}n^{-(a-1/2)}$. Reflection gives $f_{q,a}(0)=f_{1,a}(0)$, so the constant is independent of $q$. For future walks started from admissible states $(n,x_n)$ with $x_n/\\sqrt n\\to z$, we establish functional convergence of the centered martingale-normalized paths to a time-changed Brownian motion. Uniform control of late hitting times then gives the eventual-return kernel $2Φ(-\\sqrt{2a-1}|z|)$, uniformly on bounded scaled windows, where $Φ$ is the standard normal distribution function. The same last-visit asymptotic holds uniformly for nonempty deterministic targets $F_n\\subseteq[-r_n,r_n]\\cap\\mathbb{Z}$, $r_n=o(\\sqrt n)$, held fixed during each trajectory. Conditionally on $T_0\\geq n$, we identify the joint limit of $S_n/\\sqrt n$, the next return divided by $n$, and the last return divided by the next return. The last ratio converges to $A^{-1/(2a-1)}$, independently of the other two limiting coordinates, where $A$ has the arcsine distribution on $(0,1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01279",
  "author_total": 1,
  "authors": [
    {
      "name": "Hélio Trinas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cybzhcennnqt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01279",
  "fulltext_url": null,
  "id": "5cybzhcennnqt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01279",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01279",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cybzhcennnqt.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01279",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01279"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Prodigal Elephant Returns: Last-Return Laws in the Superdiffusive Regime",
  "title_text": "The Prodigal Elephant Returns: Last-Return Laws in the Superdiffusive Regime",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5cybzhcennnqt",
  "venue": "arXiv",
  "versions": []
}