Research Article · Preprint Mathematics
Dimension Thresholds for 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
- 1 · Permanent link
Overview
Abstract
For a nonnegative coordinatewise nondecreasing integrable weight on a finite d-dimensional box, joint Gamma-normalized mixed moments are log-concave when every exponent is at least d/4, for d at least two. No input log-concavity, independence or positive association is required. Finite product grids with ordered log-concave coordinate values and index-increasing weights admit the sufficient bound d/4+4. Continuous densities and finite-grid examples show that both thresholds have optimal leading constant one quarter; additive constants remain open. Conditional logarithmic-deficit estimates, a signed-block covariance argument and trigamma bounds also give spatial product-power smoothing for arbitrary decreasing inputs at powers d/4+4 in the continuous case and d/4+7 for arrays. A classical matroid input yields a dimension-free binary endpoint, while unweighted hereditary binary supports require dimension-linear powers.
Keywords Classical and Real Analysis · Probability and Statistics
Scope of the work
For a nonnegative coordinatewise nondecreasing integrable weight on a finite d-dimensional box, joint Gamma-normalized mixed moments are log-concave when every exponent is at least d/4, for d at least two. No input log-concavity, independence or positive association is required. Finite product grids with ordered log-concave coordinate values and index-increasing weights admit the sufficient bound d/4+4. Continuous densities and finite-grid examples show that both thresholds have optimal leading constant one quarter; additive constants remain open. Conditional logarithmic-deficit estimates, a signed-block covariance argument and trigamma bounds also give spatial product-power smoothing for arbitrary decreasing inputs at powers d/4+4 in the continuous case and d/4+7 for arrays. A classical matroid input yields a dimension-free binary endpoint, while unweighted hereditary binary supports require dimension-linear powers. No exact additive optimality or new scalar Mellin principle claimed. Moment-exponent theorems retain finite grids/boxes; spatial smoothing permits infinite total mass/support with local finiteness/integrability. No higher-dimensional square-kernel, general nonmonotone-input theorem, new matroid/graph principle or real-witness-to-integer-TU-Jensen counterexample claimed. Classical ancestry and Common-Martin-Stubbe full-primary access limit explicitly disclosed.
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Scott Schubr. (2026). Dimension Thresholds for Concavity of Normalized Mixed Moments (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/dimension-thresholds-for-concavity-of-normalized-mixed-moments/v/1
BibTeX
@techreport{luminary_dimension_thresholds_for_concavity_of_normalized_mixed_moments_v1,
title = {Dimension Thresholds for Concavity of Normalized Mixed Moments},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/dimension-thresholds-for-concavity-of-normalized-mixed-moments/v/1}
}Loading discussion…