Research Article · Preprint Mathematics
Moment certificates for homogeneous stable count laws
- Primary subject
- Mathematics
- Subject subfield
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · 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 = (m − 1 − Σᵢ Var(Aᵢ)/tᵢ)/2. We prove TV(P, M) ≤ min{1, CW} with C = 20/(2√10 − 1) < 3.757, 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, then uses positivity of the likelihood ratio. A classical hyperbolic-slice reduction gives finite exact coefficient certificates. A standalone rational-arithmetic verification proves the stronger bound TV(P, M) ≤ min{1, 2W} for every 1 ≤ n ≤ 50. 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
Applies to nonnegative normalized homogeneous real-stable count generating polynomials, with arbitrary degree and category count and positive means after deleting zero-mean categories. Proves the uniform total-variation coefficient 20/(2√10 − 1), and coefficient two in degrees 1–50 by an exact finite certificate. The sharp all-degree coefficient two remains open. Credits Han–Niles-Weed, Banach, orthogonal expansion methods, classical divergence bounds and hyperbolic-slice root extrema. No worldwide-priority, breakthrough, algorithmic runtime, sampling or statistical sample-complexity claim. This specialist extension is a separate paper; earlier stable-polynomial publications remain unchanged.
Reproduction notes
The eight-member package contains the complete seven-page PDF, self-contained main.tex, references.bib, verify_coefficients.py, README.md, CITATION.cff, manifest.json and SHA256SUMS. Run the standard-library Python 3.10+ verifier to check every stated finite-degree coefficient inequality through degree 50 using exact integer/Fraction arithmetic. These finite checks do not formally verify the analytic proof or establish the all-degree conjecture. The package-only source rebuild matched all seven pages’ text, dimensions and rendered pixels.
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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
- S. Banach, Über homogene Polynome in L², Studia Mathematica 7 (1938), 36–44.
- D. Carando and J. T. Rodríguez, Symmetric multilinear forms on Hilbert spaces: where do they attain their norm? (2018).
- W. Ehm, Binomial approximation to the Poisson binomial distribution, Statistics & Probability Letters 11(1) (1991), 7–16.
- A. Guntuboyina, S. Saha and G. Schiebinger, Sharp inequalities for f-divergences (2013).
- 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.
- Y. Han and J. Niles-Weed, Elementary symmetric polynomial inequalities for centered vectors and matrices (2026).
- W.-L. Loh, Stein’s method and multinomial approximation, The Annals of Applied Probability 2(3) (1992), 536–554.
- C. Riener and R. Schabert, Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions (2023), version 2.
- B. Roos, Closeness of convolutions of probability measures, Bernoulli 16(1) (2010), 23–50.
- 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 1). Luminary Research Archive. https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/1
BibTeX
@techreport{luminary_moment_certificates_homogeneous_stable_count_laws_v1,
title = {Moment certificates for homogeneous stable count laws},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/1}
}Loading discussion…