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Research Article · Preprint Mathematics

Moment certificates for homogeneous stable count laws

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Probability and Statistics
Primary subject
Mathematics
Subject subfield
Probability and Statistics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
3 · Permanent link
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Overview

Abstract

Let P be a probability law on nonnegative integer count vectors of total size n, with a homogeneous real-stable probability generating polynomial and positive mean vector t. We compare P with M = Mult(n, t/n) using the weighted covariance deficit W = ½(m − 1 − ∑ᵢ Var(Aᵢ)/tᵢ). We prove TV(P, M) ≤ min{1, (147/68)W}, independently of degree, category count and the mean vector. The argument transfers the centered elementary-symmetric-polynomial inequality of Han and Niles-Weed through Banach’s theorem and an orthogonal multinomial density expansion. It retains explicit low orders, handles exceptional discrete root profiles, and combines exact polynomial certificates with positivity of the likelihood ratio. A classical hyperbolic-slice reduction also gives the stronger bound TV(P, M) ≤ min{1, 2W} for every degree from one through fifty. Standalone rational-arithmetic code verifies all finite and continuous-domain certificates. Every all-degree linear coefficient must be at least two; whether two suffices in all degrees remains open. The results include nonfactorizable stable generating polynomials and make no algorithmic or statistical-estimation claim.

Keywords real-stable polynomials · multinomial approximation · total variation · covariance deficit · moment certificates · exact rational verification

Scope of the work

Finite-category homogeneous real-stable count laws with positive mean coordinates; zero-mean coordinates may be deleted. The all-degree total-variation coefficient is 147/68, independent of degree, category count and the mean vector. The stronger coefficient two is proved through degree fifty. Coefficient two for every degree remains open, and any universal linear coefficient must be at least two. The fixed-order scalar certificates through order twelve are uniform in multiplicity; they do not prove an all-order coefficient-one theorem. Classical Banach transfer, centered elementary-symmetric-polynomial inequalities, hyperbolic-slice reduction and orthogonal expansions are credited inputs. No algorithmic or statistical-estimation guarantee is asserted.

Reproduction notes

The nine-member publication package contains the twelve-page PDF, self-contained main.tex, optional references.bib, two exact rational-arithmetic Python verification programs, reader README, citation metadata, manifest and checksums aligned to edition three. Python 3.10 or newer with its standard library suffices for verify_coefficients.py and verify_refined.py, including optimized Python execution. Compile main.tex with Tectonic or run pdflatex twice. The exact fixed-order infinite-multiplicity polynomial certificates, exceptional profiles and global interval checks supplement the analytic proof and do not establish the open all-degree coefficient-two conjecture.

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Author disclosures

Generative AI systems assisted mathematical exploration, literature discovery, proof drafting, exact-code development and independent adversarial proof checks. These are internal AI-assisted checks, not external human refereeing. The author retains responsibility for the manuscript. Classical inputs and the close Han–Niles-Weed methodology are explicitly attributed; originality remains qualified by the inspected literature.

References and related publications

  1. S. Banach, Über homogene Polynome in L², Studia Mathematica 7 (1938), 36–44.
  2. D. Carando and J. T. Rodríguez, Symmetric multilinear forms on Hilbert spaces: where do they attain their norm? (2018).
  3. W. Ehm, Binomial approximation to the Poisson binomial distribution, Statistics & Probability Letters 11(1) (1991), 7–16.
  4. A. Guntuboyina, S. Saha and G. Schiebinger, Sharp inequalities for f-divergences (2013).
  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; coefficient result: Theorem 4.3(2), arXiv version 3.
  6. Y. Han and J. Niles-Weed, Elementary symmetric polynomial inequalities for centered vectors and matrices (2026).
  7. W.-L. Loh, Stein’s method and multinomial approximation, The Annals of Applied Probability 2(3) (1992), 536–554.
  8. C. Riener and R. Schabert, Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions (2023), version 2.
  9. B. Roos, Closeness of convolutions of probability measures, Bernoulli 16(1) (2010), 23–50.
  10. B. Roos, On Bobkov’s approximate de Finetti representation via approximation of permanents of complex rectangular matrices (2014), version 2.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Moment certificates for homogeneous stable count laws (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/3

BibTeX
@techreport{luminary_moment_certificates_homogeneous_stable_count_laws_v3,
  title = {Moment certificates for homogeneous stable count laws},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 3},
  url = {https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/3}
}

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