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  "slug": "separated-rates-exact-polya-order",
  "title": "Exact total positivity from separated exponential rates: a matrix dimension theorem and the q-Gindikin conjecture",
  "shortTitle": "Slow stages fix the exact Pólya frequency order",
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  "oneLine": "A waiting time of m slow exponential stages and one fast stage of random speed has Pólya frequency order exactly m+1, never more, and m+1 is also its largest matrix-lift dimension; this proves the parameter region of Sra's q-Gindikin Conjecture 6.1.",
  "abstract": "A Pólya frequency (PF) density of order r is one whose shift matrices [f(x_i − y_j)] of size up to r all have nonnegative determinants: order two is log-concavity, and higher orders are stronger forms of variation-diminishing smoothing. A separate question asks whether a scalar Laplace transform φ lifts to a random d×d positive semidefinite complex Hermitian matrix with transform ∏φ(λ_i(S)), a matrix analogue of gamma and Wishart laws; Sra (2026, version 1 of 29 October, Conjecture 6.1) needed exactly this for a q-product. This candidate studies a waiting time made of m exponential stages with distinct slow rates a_1 < … < a_m followed by one fast stage whose rate is drawn from an arbitrary probability measure on [b_0, ∞), with b_0 > a_m. Theorem 1: the density is PF of order m+1; if the fast rate is genuinely random, the exact order is m+1, the next derivative Wronskian W_{m+2} is strictly negative at every x > 0, and a matrix lift exists if and only if d ≤ m+1. A deterministic fast rate gives the classical PF∞ case with lifts in every dimension. No moments of the mixing measure are needed. As an application the paper proves the complete parameter region of Sra's Conjecture 6.1: for 0 < q < 1 and 0 ≤ t < 1, a dimension-d law exists if and only if t ∈ ([0, q^{d−1}] ∪ {q, …, q^{d−2}}) ∩ [0, 1); Sra had proved necessity, the geometric points and d = 2. Section 6A extends the exact conclusions to the weak gap μ((a_m, ∞)) = 1 and, separately, to repeated slow rates, and gives real symmetric lifts in every dimension d ≤ m+1; an interleaved example shows that some rate ordering is necessary. The distribution class and the scalar tools are classical: the finite-mixture case lies in the class SCM of Sumita and Masuda (1987), and in their CMPF_n exactly when m = r + 1 for r mixture atoms. That paper's only Pólya-frequency-type property is classical log-concavity, with no higher orders, exactness or matrix lifts, so it does not contain Theorem 1; the sharp theorem was not found in the other sources checked either. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
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  "keywords": [
    "total positivity",
    "Pólya frequency functions",
    "exponential mixtures",
    "phase-type distributions",
    "Coxian distributions",
    "Wishart laws",
    "matrix lifts",
    "q-Gindikin",
    "Macdonald (q,t)-permanent",
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    "Exact order (Theorem 1): with distinct slow rates a_1 < … < a_m below every possible fast rate (b_0 > a_m), the density is PF of order m+1; if the fast rate is genuinely random, W_{m+2} < 0 at every x > 0, so the exact order is m+1, never more.",
    "Matrix dimension (Theorem 1): a complex Hermitian matrix lift with transform ∏φ(λ_i(S)) exists if and only if d ≤ m+1, with a continuous, strictly positive density on the positive definite cone; a deterministic fast rate gives the classical PF∞ case with lifts in every dimension.",
    "q-Gindikin (Theorem 4): for 0 < q < 1 and 0 ≤ t < 1, a dimension-d matrix law exists if and only if t ∈ ([0, q^{d−1}] ∪ {q, …, q^{d−2}}) ∩ [0, 1), proving Sra's Conjecture 6.1; the new part is sufficiency in the continuous range for d ≥ 3.",
    "Extensions (Section 6A): the same exact conclusions under the weak gap μ((a_m, ∞)) = 1 and, separately, for repeated slow rates; closure under limits; real symmetric lifts in every dimension d ≤ m+1.",
    "Ordering matters (Section 6): E_2 ∗ (½E_1 + ½E_3) has W_2(log 8) = −13/16384 < 0, so it is not even log-concave."
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  "relatedWorks": [
    {
      "citation": "Sumita and Masuda (1987). Classes of probability density functions having Laplace transforms with negative zeros and poles. Advances in Applied Probability 19(3). Introduces the classes CMPF_n and SCM; the finite-mixture case studied here lies in SCM (in CMPF_n only when m = r + 1). Read in full: not prior work for Theorem 1.",
      "url": "https://doi.org/10.2307/1427410",
      "doi": "10.2307/1427410"
    },
    {
      "citation": "Schoenberg (1951). On Pólya frequency functions I: the totally positive functions and their Laplace transforms. Journal d'Analyse Mathématique. Characterises PF∞ densities; Remark 6A.8 rests on this representation.",
      "url": "https://doi.org/10.1007/BF02790092",
      "doi": "10.1007/BF02790092"
    },
    {
      "citation": "Assaf, Shaked and Shanthikumar (1985). First-passage times with PFr densities. Journal of Applied Probability. Sufficient conditions for finite PF order with interlacing poles and zeros; no exact order or Wronskian sign is given there.",
      "url": "https://doi.org/10.2307/3213758",
      "doi": "10.2307/3213758"
    },
    {
      "citation": "Förster, Kieburg and Kösters (2021). Polynomial ensembles and Pólya frequency functions. Journal of Theoretical Probability. Establishes the correspondence between finite-order PF functions and derivative-type matrix ensembles used here.",
      "url": "https://doi.org/10.1007/s10959-020-01030-z",
      "doi": "10.1007/s10959-020-01030-z"
    },
    {
      "citation": "Kieburg and Zhang (2023). Derivative principles for invariant ensembles. Advances in Mathematics. The standard route from such ensembles to scalar derivative determinants.",
      "url": "https://doi.org/10.1016/j.aim.2022.108833",
      "doi": "10.1016/j.aim.2022.108833"
    },
    {
      "citation": "Belton, Guillot, Khare and Putinar (2022). Hirschman–Widder densities. Applied and Computational Harmonic Analysis. Total-positivity properties of finite exponential convolutions.",
      "url": "https://doi.org/10.1016/j.acha.2022.04.002",
      "doi": "10.1016/j.acha.2022.04.002"
    },
    {
      "citation": "Bertoin, Biane and Yor (2004). Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions. Source of the scalar q-product law; no novelty is claimed for that law.",
      "url": "https://doi.org/10.1007/978-3-0348-7943-9_3",
      "doi": "10.1007/978-3-0348-7943-9_3"
    },
    {
      "citation": "Faraut and Korányi (1994). Analysis on symmetric cones. Oxford University Press. Background on Laplace transforms on the positive semidefinite cone.",
      "url": "https://doi.org/10.1093/oso/9780198534778.001.0001",
      "doi": "10.1093/oso/9780198534778.001.0001"
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