{
  "abstract": "Bernard and Letac (1971) introduced a method for uniform random sampling among $m$ outcomes from an unknown biased source of independent and identically distributed symbols. The process terminates when the multinomial coefficient of the cumulative symbol counts equals zero modulo $m$. This study extends the computational and information-theoretic analysis of their construction by presenting five algorithms with formal correctness guarantees and comprehensive complexity analyses. For prime $m=p$, the Bernard-Letac framework is analyzed in greater detail. The Rényi entropies of the source yield an exact product formula for the expected number of draws. A first-order approximation consistently overestimates this value, and the entropy lower bound is never attained. As $p$, treated as a continuous parameter, approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, for every prime modulus $p$ and every finite alphabet $I$, an explicit automaton with $p + |I| + 3$ states computes the mod-$p$ first-passage kernel of the walk from the base-$p$ digits of its arguments, reducing the fair assignment cost from quadratic to nearly linear.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.20234",
  "author_total": 1,
  "authors": [
    {
      "name": "Claude Gravel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71cf5vq6zxphq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.20234",
  "fulltext_url": null,
  "id": "71cf5vq6zxphq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.20234",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7203894144",
  "pdf_url": "https://arxiv.org/pdf/2608.20234",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71cf5vq6zxphq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2608.20234",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.20234"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction",
  "title_text": "Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/71cf5vq6zxphq",
  "venue": "arXiv",
  "versions": []
}