Dimension Thresholds for Concavity of Normalized Mixed Moments
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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. The theorem applies to arbitrary inputs in its stated domain. 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. 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.
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Research details
- Subjects
- Mathematics · Probability and Statistics · Classical and Real Analysis
- Keywords
- Classical and Real Analysis, Probability and Statistics
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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. The theorem applies to arbitrary inputs in its stated domain. 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. 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. Moment-exponent theorems retain finite grids/boxes; spatial smoothing permits infinite total mass/support with local finiteness/integrability. Classical ancestry and Common-Martin-Stubbe full-primary access limit explicitly disclosed.
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AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work.
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