{
  "abstract": "Let $B_1,\\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\\ldots,p_n$, and let $S=\\sum_i B_i$. Hillion and Johnson proved that the Shannon entropy of $S$ is jointly concave in the parameter vector and proposed corresponding critical-order conjectures for R'enyi and Tsallis entropies, with predicted thresholds $2$ and approximately $3.65986$, respectively. We determine both thresholds exactly. For every $0<q<1$, the power sum $\\sum_k \\mathbb P(S=k)^q$ is jointly concave in $(p_1,\\ldots,p_n)$, and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order $q$ are jointly concave. At $q=1$ this agrees with the Shannon theorem. For every $q>1$, joint concavity fails already for the sum of two Bernoulli variables: a transverse interpolation in which the two parameters move in opposite directions gives strict local convexity for both entropies. Hence the universal joint-concavity range for both families is exactly $0<q\\leq 1$. Below order one, the proof combines the Hillion--Johnson transport inequality with an explicit nonlinear telescoping correction. The corrected local curvature reduces to a two-dimensional quadratic form. An exact Riccati identity, together with a one-sided zero-crossing argument, proves positivity of its determinant throughout the full range $0<q<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27433",
  "author_total": 1,
  "authors": [
    {
      "name": "Haoran Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0w8m71x0k4v1c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27433",
  "fulltext_url": null,
  "id": "0w8m71x0k4v1c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27433",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7214171386",
  "pdf_url": "https://arxiv.org/pdf/2609.27433",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0w8m71x0k4v1c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27433",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27433"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem",
  "title_text": "The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/0w8m71x0k4v1c",
  "venue": "arXiv",
  "versions": []
}