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The Gaussian product inequality is an important conjecture concerning themoments of Gaussian random vectors. While all attempts to prove the Gaussianproduct inequality in full generality have been unsuccessful to date, numerouspartial results have been derived in recent decades and we provide here furtherresults on the problem. Most importantly, we establish a strong version of theGaussian product inequality for multivariate gamma distributions in the case ofnonnegative correlations, thereby extending a result recently derived by Genestand Ouimet [5]. Further, we show that the Gaussian product inequality holdswith nonnegative exponents for all random vectors with positive componentswhenever the underlying vector is positively upper orthant dependent. Finally,we show that the Gaussian product inequality with negative exponents followsdirectly from the Gaussian correlation inequality.

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions