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  "slug": "gamma-sum-sharp-moments",
  "title": "Sharp moments of nonnegative gamma sums on the Euclidean sphere",
  "shortTitle": "Spread or concentrate? The sharp moments of weighted gamma sums",
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  "oneLine": "For a weighted sum of independent exponential or gamma waiting times with its variance fixed, the weights that make the q-th moment smallest climb a staircase (one weight up to q = 5.32, then two, three, … equal weights), proving the Brazitikos–Pandis conjecture for every real q; the largest moment is found too, and for three or more variables the best weights jump.",
  "abstract": "Let X₁, …, Xₙ be independent gamma variables with common shape α > 0 and unit rate, and let S = Σaᵢ Xᵢ with aᵢ ≥ 0 and Σaᵢ² = 1, so that the variance of S is fixed. For every real q > 0 the paper determines both the minimum and the maximum of E Sᵠ. The minimum equals min over 1 ≤ m ≤ n of Γ(mα+q)/(Γ(mα) m^{q/2}) and is attained only at vectors with m equal nonzero coefficients; for exponential variables one coordinate is optimal up to q = 5.3197…, then two, then three, the optimal number growing one at a time and roughly like 0.40 q. For α = 1 this is Conjecture 4.8 of Brazitikos and Pandis, previously known for integer q, for q ≤ 4 (Brazitikos, Tang and Tkocz) and, in an unrefereed AI-generated dossier, for q ≤ 5. The maximum is attained at the flat vector or at a vector with one larger coefficient and n−1 equal smaller ones, located by the unique interior local maximum of an explicit hypergeometric function; the flat vector is the unique maximiser below a switch exponent q*, with q* = 2α+2 and a continuous switch for n = 2, and 2+2α√(n−1) < q* < nα+2 with a jump for n ≥ 3. The proofs combine gamma size bias, a second-order exclusion argument and an exact sign formula for the Gauss hypergeometric equation. The paper also proves the large-exponent asymptotics of the minimum, settles negative exponents, and records consequences for positive Gaussian quadratic forms and symmetric Dirichlet averages. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-03",
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  "keywords": [
    "gamma distribution",
    "sums of independent random variables",
    "sharp moment inequalities",
    "exponential random variables",
    "Brazitikos–Pandis conjecture",
    "Gaussian quadratic forms",
    "hypergeometric function",
    "Khinchin-type inequalities"
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    "Sharp minimum (Theorem 1, proof): for every n, every gamma shape α > 0 and every real q > 0, min E(Σaᵢ Xᵢ)ᵠ over nonnegative unit vectors equals min over m of Γ(mα+q)/(Γ(mα) m^{q/2}), attained exactly at m equal nonzero weights. For α = 1 this proves Brazitikos–Pandis Conjecture 4.8.",
    "Staircase (Section 6 and Proposition 4, proof): the optimal number of equal weights rises by one at a time; for exponentials it is 1 up to q = 5.3197…, 2 up to 7.88, 3 up to 10.41, and grows like q/2.5129 ≈ 0.40 q.",
    "Sharp maximum (Theorem 3, proof): the maximiser is the flat vector or has one larger weight and the rest equal, located by the unique interior local maximum of an explicit hypergeometric function; flat wins below a switch q*. For n = 2, q* = 2α + 2 and the switch is continuous; for n ≥ 3, 2 + 2α√(n−1) < q* < nα + 2 and the maximiser jumps (three exponentials: q* ≈ 4.8535, numerical).",
    "Consequences (Sections 7 and 9, proof): sharp constants for positive Gaussian quadratic forms (at q = 4 the minimum 96 is at rank two, against 105 at rank one), symmetric Dirichlet averages, and negative exponents."
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  "relatedWorks": [
    {
      "citation": "Brazitikos and Pandis (2025). Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables. arXiv. Source of the real-exponent conjecture (Conjecture 4.8) proved here; integer exponents, the integer maximiser shape and two variables.",
      "url": "https://arxiv.org/abs/2512.12254",
      "doi": "10.48550/arXiv.2512.12254"
    },
    {
      "citation": "Brazitikos, Tang and Tkocz (2026). Moments of sums of exponentials, beyond CHS. arXiv. The case q ≤ 4 and the size-bias and critical-point identities used throughout.",
      "url": "https://arxiv.org/abs/2602.03058",
      "doi": "10.48550/arXiv.2602.03058"
    },
    {
      "citation": "Korea Superintelligence Labs, generated by Machina Mathematica (2026). The h4 inequality behind Hunter's conjecture: an exact real relaxation, two errata, and the real-exponent conjecture of Brazitikos and Pandis. AI-generated research dossier, version 1. The case q ≤ 5, the real-q two-value reduction, a conjectured curvature claim and the residual window closed here.",
      "url": "https://ideosphere.ai/papers/brazitikos-realq",
      "doi": null
    },
    {
      "citation": "Chasapis, Singh and Tkocz (2023). Entropies of Sums of Independent Gamma Random Variables. Journal of Theoretical Probability. Schur-convexity result giving the location of both extremes for negative exponents.",
      "url": "https://doi.org/10.1007/s10959-022-01192-y",
      "doi": "10.1007/s10959-022-01192-y"
    },
    {
      "citation": "Borell (1973). Complements of Lyapunov's inequality. Mathematische Annalen. Log-concavity of moments, behind the low-exponent results.",
      "url": "https://doi.org/10.1007/BF01362702",
      "doi": "10.1007/BF01362702"
    },
    {
      "citation": "Hunter (1977). The positive-definiteness of the complete symmetric functions of even order. Mathematical Proceedings of the Cambridge Philosophical Society. Origin of the line of work on moments of exponential sums.",
      "url": "https://doi.org/10.1017/S030500410005386X",
      "doi": "10.1017/S030500410005386X"
    },
    {
      "citation": "Haagerup (1981). The best constants in the Khintchine inequality. Studia Mathematica. The classical sharp-constant problem for Rademacher sums.",
      "url": "https://doi.org/10.4064/sm-70-3-231-283",
      "doi": "10.4064/sm-70-3-231-283"
    },
    {
      "citation": "Latała and Oleszkiewicz (1995). A note on sums of independent uniformly distributed random variables. Colloquium Mathematicum. Sharp moment comparison for uniform summands.",
      "url": "https://doi.org/10.4064/cm-68-2-197-206",
      "doi": "10.4064/cm-68-2-197-206"
    },
    {
      "citation": "Nayar and Oleszkiewicz (2012). Khinchine type inequalities with optimal constants via ultra log-concavity. Positivity. Sharp constants for symmetric discrete summands.",
      "url": "https://doi.org/10.1007/s11117-011-0130-z",
      "doi": "10.1007/s11117-011-0130-z"
    },
    {
      "citation": "Havrilla and Tkocz (2021). Sharp Khinchin-type inequalities for symmetric discrete uniform random variables. Israel Journal of Mathematics. Further sharp Khinchin-type constants.",
      "url": "https://doi.org/10.1007/s11856-021-2244-8",
      "doi": "10.1007/s11856-021-2244-8"
    },
    {
      "citation": "Eskenazis, Nayar and Tkocz (2018). Sharp comparison of moments and the log-concave moment problem. Advances in Mathematics. Sharp moment comparison for log-concave summands.",
      "url": "https://doi.org/10.1016/j.aim.2018.06.014",
      "doi": "10.1016/j.aim.2018.06.014"
    },
    {
      "citation": "Gluskin and Kwapień (1995). Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Mathematica. Moment estimates for sums with log-concave tails.",
      "url": "https://doi.org/10.4064/sm-114-3-303-309",
      "doi": "10.4064/sm-114-3-303-309"
    },
    {
      "citation": "Latała (1997). Estimation of moments of sums of independent real random variables. The Annals of Probability. Two-sided moment estimates for general summands.",
      "url": "https://doi.org/10.1214/aop/1024404522",
      "doi": "10.1214/aop/1024404522"
    },
    {
      "citation": "Bartczak, Nayar and Zwara (2021). Sharp Variance-Entropy Comparison for Nonnegative Gaussian Quadratic Forms. IEEE Transactions on Information Theory. Extremal problems for positive Gaussian quadratic forms (the case α = 1/2).",
      "url": "https://doi.org/10.1109/TIT.2021.3113281",
      "doi": "10.1109/TIT.2021.3113281"
    },
    {
      "citation": "Székely and Bakirov (2003). Extremal probabilities for Gaussian quadratic forms. Probability Theory and Related Fields. Extremal tail probabilities for the same forms.",
      "url": "https://doi.org/10.1007/s00440-003-0262-6",
      "doi": "10.1007/s00440-003-0262-6"
    },
    {
      "citation": "Pang (2026). A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms. arXiv. Related sharp moment bounds for Gaussian quadratic forms.",
      "url": "https://arxiv.org/abs/2609.05914",
      "doi": "10.48550/arXiv.2609.05914"
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