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

TL;DR

This paper advances the understanding of the Gaussian product inequality by establishing new results for multivariate gamma distributions with nonnegative correlations and for positively upper orthant dependent vectors, linking to the Gaussian correlation inequality.

Contribution

It provides new partial results on the Gaussian product inequality, including a strong version for multivariate gamma distributions and conditions for positive and negative exponents.

Findings

Established a strong Gaussian product inequality for multivariate gamma distributions with nonnegative correlations.

Proved the inequality holds for positively upper orthant dependent vectors with nonnegative exponents.

Showed that the inequality with negative exponents follows from the Gaussian correlation inequality.

Abstract

The Gaussian product inequality is an important conjecture concerning the moments of Gaussian random vectors. While all attempts to prove the Gaussian product inequality in full generality have been unsuccessful to date, numerous partial results have been derived in recent decades and we provide here further results on the problem. Most importantly, we establish a strong version of the Gaussian product inequality for multivariate gamma distributions in the case of nonnegative correlations, thereby extending a result recently derived by Genest and Ouimet [5]. Further, we show that the Gaussian product inequality holds with nonnegative exponents for all random vectors with positive components whenever the underlying vector is positively upper orthant dependent. Finally, we show that the Gaussian product inequality with negative exponents follows directly from the Gaussian correlation inequality.