Research Article · Preprint Mathematics
Dimension thresholds for log-concavity of normalized mixed moments
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis · Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
For a nonnegative coordinatewise nondecreasing integrable weight on a finite d-dimensional box, we prove joint log-concavity of its mixed moments divided by ∏_iΓ(α_i+1) whenever α_i≥ d/4, d≥2. The weight need not be log-concave, independent across coordinates, or positively associated. For finite product grids with ordered log-concave coordinate values and a weight nondecreasing in the coordinate indices, the corresponding sufficient orthant is α_i≥ d/4+4. Increasing upper-corner densities and finite-grid weights give counterexamples at common exponent d/4-O(1). Thus the universal dimension thresholds in the two classes both have optimal leading constant 1/4. The proof combines conditional logarithmic-deficit bounds with a signed-block covariance estimate and elementary trigamma inequalities. Explicit rational finite-grid and continuous counterexamples accompany the asymptotic construction. The same method gives spatial product-power smoothing of arbitrary decreasing densities and arrays at powers d/4+4 and d/4+7, respectively, again with optimal leading coefficient 1/4. A classical matroid log-concavity input yields a dimension-free binary endpoint, whereas unweighted hereditary binary supports already require dimension-linear powers.
Keywords Classical and Real Analysis · Probability and Statistics
Scope of the work
Joint log-concavity of Gamma-normalized mixed moments and spatial power smoothing for arbitrary monotone inputs, with optimal leading dimension coefficient 1/4 and distinct binary-structure thresholds.
Reproduction notes
15-page manuscript and 6 package members. Compile main.tex with compatible LaTeX or Tectonic. Run python -B verify_moments.py and python -O -B verify_moments.py using the Python 3 standard library; both print the same exact rational witness and polynomial-certificate results without reading external data or writing files.
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Author disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Dimension thresholds for log-concavity of normalized mixed moments (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/dimension-thresholds-for-concavity-of-normalized-mixed-moments/v/2
BibTeX
@techreport{luminary_dimension_thresholds_for_concavity_of_normalized_mixed_moments_v2,
title = {Dimension thresholds for log-concavity of normalized mixed moments},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/dimension-thresholds-for-concavity-of-normalized-mixed-moments/v/2}
}Loading discussion…