Dimension thresholds for log-concavity of normalized mixed moments

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.