{
  "abstract": "We study the minimal random walk with independent positive random step sizes. By separating the movement count from the centered step-size noise, we obtain normal approximations and self-normalized Cramér-type moderate deviations under finite-moment assumptions. For the noise, conditioning on the movement count yields Kolmogorov, Wasserstein and Zolotarev bounds. A polynomial moderate-deviation range holds when a recalled stop can lead to movement. At the boundary where this probability is zero, we obtain a logarithmic range and construct a bounded nonsymmetric example excluding a general polynomial range. For the full position, a weighted martingale representation gives limit laws in the different regimes, nonuniform normal approximations and moment-distance bounds without logarithmic losses. Variance concentration also yields self-normalized moderate deviations in the diffusive and critical regimes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29156",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuyuan Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bp2j341y1heqh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29156",
  "fulltext_url": null,
  "id": "bp2j341y1heqh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29156",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7223205014",
  "pdf_url": "https://arxiv.org/pdf/2610.29156",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bp2j341y1heqh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29156",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29156"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes",
  "title_text": "Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/bp2j341y1heqh",
  "venue": "arXiv",
  "versions": []
}