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

Scope

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. Any universal linear coefficient must be at least two. The fixed-order scalar certificates through order twelve are uniform in multiplicity. Classical Banach transfer, centered elementary-symmetric-polynomial inequalities, hyperbolic-slice reduction and orthogonal expansions are credited inputs.

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 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.

References and related resources

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