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  "slug": "stable-power-concavity-gaussian-boundary",
  "title": "Power concavity of symmetric stable densities: sharp degeneration at the Gaussian boundary",
  "shortTitle": "A shape constraint breaks near the Gaussian",
  "url": "https://evidencepress.org/releases/stable-power-concavity-gaussian-boundary/",
  "oneLine": "Symmetric stable densities approach a Gaussian locally while their best global power-concavity exponent deteriorates logarithmically.",
  "abstract": "A written proof candidate gives the sharp Gaussian-boundary asymptotic for the optimal density power-concavity exponent: minus one sixth times log(1/(2-alpha)). A separate tail argument excludes the proposed exponent -1/(1+alpha) at every non-Gaussian symmetric stable index except Cauchy, contradicting the density conjecture in the inspected Laha–Miao–Wellner arXiv version. An interval certificate supplies a rational witness outside the density range [-1,0]. Weaker distribution-function constraints and historical priority remain separate questions.",
  "datePublished": "2026-09-08",
  "dateModified": "2026-09-08",
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  "keywords": [
    "stable densities",
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    "Strict tail-exponent obstruction for every non-Gaussian non-Cauchy symmetric stable law.",
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    {
      "citation": "Laha, Miao and Wellner, Bi-s*-concave distributions (2021); density conjecture inspected in arXiv v2 §5 p.21.",
      "url": "https://arxiv.org/abs/2006.03989v2",
      "doi": "10.48550/arXiv.2006.03989"
    },
    {
      "citation": "Doss and Wellner, Inference for the mode of a log-concave density (2019), §6.2.2.",
      "url": "https://doi.org/10.1214/18-AOS1770",
      "doi": "10.1214/18-AOS1770"
    },
    {
      "citation": "Bobkov and Madiman, The entropy per coordinate of a random vector is highly constrained under convexity conditions (2011).",
      "url": "https://arxiv.org/abs/1006.2883",
      "doi": "10.48550/arXiv.1006.2883"
    },
    {
      "citation": "Matsui and Takemura, Some improvements in numerical evaluation of symmetric stable density and its derivatives (2006).",
      "url": "https://doi.org/10.1080/03610920500439729",
      "doi": "10.1080/03610920500439729"
    }
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