Luminary Research Archive
Luminary Research ArchiveIndependent scholarship
Luminary Portal
PublicEdition 8

Covariance-Spectrum Certificates for Homogeneous Stable Count Laws

Scott Schubr ORCID

Original PDF · First-page preview

Loading original first page…

Abstract

We compare a homogeneous real-stable count law with the multinomial distribution having the same mean. The normalized covariance deficit determines the quadratic orthogonal component of the likelihood ratio and controls its higher-order remainder through explicit spectral bounds. These estimates give a uniform quadratic approximation in chi-square distance as the trace deficit vanishes, and convergence for diffuse covariance-deficit spectra with bounded trace, allowing the degree and number of categories to vary. They also yield a sharp covariance-matching exponent, while nonfactorizable stable cubics show why a stronger Frobenius-power estimate for categorical products cannot extend to the full stable class. An optimal all-degree centered-real elementary-symmetric inequality, with equality classification, supplies the coefficient input and gives an optimal linear total-variation certificate. Separately, a sharp centered-complex inequality in six coordinates, including its complete equality orbit, yields the optimal uniform factor 3/4 in the complete-homogeneous bound for the squared L² norm of the fifth likelihood component of degree-six stable laws. Ternary determinantal representations transfer further complete-homogeneous coefficient bounds, and a fixed-law convolution theorem identifies the limiting Gaussian total variation. The arguments adapt classical tensor, polynomial and orthogonal-expansion methods; standalone exact-arithmetic code reconstructs the finite certificates used in the proofs.

Read the full paper in the original PDF.

Edition files

Available editions

  1. Edition 8 · 2026-10-30 · public
  2. Edition 7 · 2026-10-27 · public
  3. Edition 6 · 2026-10-24 · public
  4. Edition 5 · 2026-10-23 · public
  5. Edition 4 · 2026-10-23 · public
  6. Edition 3 · 2026-10-23 · public
  7. Edition 2 · 2026-10-23 · public
  8. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Probability and Statistics
Keywords
real-stable polynomials, covariance spectrum, multinomial approximation, total variation, elementary symmetric polynomials, exact certificates
Manuscript date
2026-10-23
Reader reuse
none

Scope

Covariance-spectrum likelihood certificates and sharp covariance-matching stability for homogeneous stable count laws, with sharp real and complex coefficient bounds, complete complex-six equality classification, full-stable cubic obstructions and fixed-law Gaussian limits.

Disclosures

OpenAI ChatGPT and Codex assisted mathematical exploration, literature comparison, drafting, exact-code development and proof checks. The author assumes responsibility for the manuscript and its mathematical claims.

Reproduction

The 41-page searchable manuscript is accompanied by complete editable sources, reader instructions, citation metadata and integrity checksums. The manuscript supplies the analytic arguments; accompanying computational materials, where included, check their specified finite certificates.

References and related resources

  1. S. Banach, Über homogene Polynome in L2, Studia Mathematica 7 (1938), 36–44. doi:10.4064/sm-7-1-36-44.
  2. D. Carando and J. T. Rodríguez, Symmetric multilinear forms on Hilbert spaces: where do they attain their norm? (2018). arXiv:1810.09373.
  3. W. Ehm, Binomial approximation to the Poisson binomial distribution, Statistics & Probability Letters 11(1) (1991), 7–16. doi:10.1016/0167-7152(91)90170-V.
  4. A. Guntuboyina, S. Saha and G. Schiebinger, Sharp inequalities for f -divergences (2013). arXiv:1302.0336.
  5. Y. Han and J. Niles-Weed, Approximate independence of permutation mixtures, The Annals of Statistics 54(3) (2026), 1594–1621. doi:10.1214/26-AOS2619. arXiv:2408.09341; the coefficient result used here is Theorem 4.3(2) in version 3.
  6. Y. Han and J. Niles-Weed, Elementary symmetric polynomial inequalities for centered vectors and matrices (2026). arXiv:2607.23836.
  7. W.-L. Loh, Stein’s method and multinomial approximation, The Annals of Applied Probability 2(3) (1992), 536–554. doi:10.1214/aoap/1177005648.
  8. C. Riener and R. Schabert, Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions (2023). arXiv:2203.08727, version 2.
  9. B. Roos, Closeness of convolutions of probability measures, Bernoulli 16(1) (2010), 23–50. doi:10.3150/08-BEJ171. arXiv:0811.0622.
  10. B. Roos, On Bobkov’s approximate de Finetti representation via approximation of permanents of complex rectangular matrices (2014). arXiv:1203.2871, version 2.
  11. P. Brändén, Polynomials with the half-plane property and matroid theory, Advances in Mathematics 216 (2007), 302–320. arXiv:math/0605678, version 4.
  12. Y.-B. Choe, J. G. Oxley, A. D. Sokal and D. G. Wagner, Homogeneous multivariate polynomials with the half-plane property, Advances in Applied Mathematics 32 (2004), 88–187. doi:10.1016/S0196-8858(03)00078-2. arXiv:math/0202034.
  13. E. Randles, Local limit theorems for complex functions on Zd, Journal of Mathematical Analysis and Applications 519 (2023), 126832. doi:10.1016/j.jmaa.2022.126832. arXiv:2201.01319, version 2.
  14. H. Derksen, P. Ivanov, C. H. Lee and E. Viola, Pseudorandomness, Symmetry, Smoothing: I, in 39th Computational Complexity Conference , LIPIcs 300 (2024), 18:1–18:27. doi:10.4230/LIPIcs.CCC.2024.18.
  15. J.-T. Hsieh, D. M. Kane, P. K. Kothari, J. Li, S. Mohanty and S. Tiegel, Rigorous Implications of the Low-Degree Heuristic (2026). arXiv:2601.05850.
  16. National Institute of Standards and Technology, Digital Library of Mathematical Functions , §18.19 and §18.23, Table 18.19.1 and equation 18.23.3.
  17. B. Roos, Multinomial and Krawtchouk approximations to the generalized multinomial distribution, Theory of Probability and its Applications 46(1) (2002),103–117. doi:10.1137/S0040585X97978750. Russian original: Teor. Veroyatnost. i Primenen. 46(1) (2001),117–133.
  18. P. Diaconis and R. C. Griffiths, Reproducing kernel orthogonal polynomials on the multinomial distribution, Journal of Approximation Theory 242 (2019),1–30. doi:10.1016/j.jat.2019.01.007. arXiv:1802.07122.
  19. J. M. Aldaz, Self-improvement of the inequality between arithmetic and geometric means, Journal of Mathematical Inequalities 3(2) (2009),213–216. arXiv:0807.1788.
  20. H. H. Bauschke, O. Güler, A. S. Lewis and H. S. Sendov, Hyperbolic polynomials and convex analysis, Canadian Journal of Mathematics 53(3) (2001),470–488. Author manuscript.
  21. S. Cao, S. He, Z. Li and Z. Wang, Extreme ratio between spectral and Frobenius norms of nonnegative tensors, SIAM Journal on Matrix Analysis and Applications 44(2) (2023),919–944. doi:10.1137/22M1502951. Author manuscript,Theorem 4.6 and Corollary 4.9.
  22. C. Hanselka, Definite determinantal representations of ternary hyperbolic forms, arXiv:1411.1661v2 (2015), Theorem 1 (Helton–Vinnikov).