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Moment certificates for homogeneous stable count laws

Scott Schubr

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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. The results include nonfactorizable stable generating polynomials.

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  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
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  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, multinomial approximation, total variation, covariance deficit, moment certificates, exact rational verification
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

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. Credits Han–Niles-Weed, Banach, orthogonal expansion methods, classical divergence bounds and hyperbolic-slice root extrema. This specialist extension is a separate paper; earlier stable-polynomial publications remain unchanged.

Disclosures

Generative AI systems assisted mathematical exploration, literature discovery, proof drafting, exact-code development and independent adversarial proof checks. The author retains responsibility for the manuscript. Classical inputs and the close Han–Niles-Weed methodology are explicitly attributed.

Reproduction

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. The supplied finite checks support the analytic proof. The package-only source rebuild matched all seven pages’ text, dimensions and rendered pixels.

References and related resources

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