{
  "abstract": "We estimate the Lévy distance to the symmetric Bernoulli distribution for normalized Boolean convolution powers of a probability measure with mean zero and variance one. If the absolute moment of order $r$ is finite, we obtain the upper bounds $o(n^{-(r-2)/6})$ for $2<r<5$ and $O(n^{-1/2})$ for $r\\ge5$. The proof relies on an inequality that controls the Lévy distance to the Bernoulli law by the values of the reciprocal Cauchy transform at two points. For each $r\\ge2$, we also construct a single fixed distribution whose absolute moments are finite exactly up to order $r$, and show that the exponent $\\min\\{(r-2)/6,1/2\\}$ cannot be improved. Under only a finite second moment assumption, the convergence can be arbitrarily slow.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39743",
  "author_total": 2,
  "authors": [
    {
      "name": "Maejima Makoto"
    },
    {
      "name": "Noriyoshi Sakuma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7wvmq4d9xc7gq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39743",
  "fulltext_url": null,
  "id": "7wvmq4d9xc7gq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39743",
  "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.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39743",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7wvmq4d9xc7gq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39743",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39743"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Berry--Esseen estimate in the Boolean central limit theorem",
  "title_text": "The Berry--Esseen estimate in the Boolean central limit theorem",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/7wvmq4d9xc7gq",
  "venue": "arXiv",
  "versions": []
}