{
  "abstract": "Let \\((S_n)_{n\\ge0}\\) be a Bernoulli random walk with \\(P(X_i=1)=p\\) and \\(P(X_i=-1)=q=1-p\\), and let \\(M_n=\\max\\{S_0,\\ldots,S_n\\}\\). We consider the semi-best selection problem of maximizing \\(P(M_N-S_τ\\le1)\\) over stopping times \\(τ\\le N\\). Allaart showed that the optimal decision in the critical state \\((n,1)\\) is determined by critical probabilities \\(p_n\\), and conjectured that \\(p_n\\to1/2\\). In this note, we prove this conjecture. For every fixed \\(p<1/2\\), the stopping value converges to \\(ν=1-(p/q)^2\\), while the continuation value converges to \\(2pν\\). Since \\(2pν<ν\\), stopping is optimal in state \\((n,1)\\) for all sufficiently large \\(n\\). Together with the known bound \\(p_n<1/2\\) for \\(n\\ge4\\), this yields \\(\\lim_{n\\to\\infty}p_n=1/2\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26670",
  "author_total": 1,
  "authors": [
    {
      "name": "José Ángel Islas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2efdntbv8gk22.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26670",
  "fulltext_url": null,
  "id": "2efdntbv8gk22",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26670",
  "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": "W7223021936",
  "pdf_url": "https://arxiv.org/pdf/2610.26670",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2efdntbv8gk22.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.26670",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26670"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convergence of the critical probabilities for stopping near the top of a random walk",
  "title_text": "Convergence of the critical probabilities for stopping near the top of a random walk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/2efdntbv8gk22",
  "venue": "arXiv",
  "versions": []
}