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

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.

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

“…This result was recovered by Edelmann et al [5] using the multivariate gamma extension of the Gaussian correlation inequality, due to Royen [22]. These authors pointed out that the component-wise absolute negative powers of multivariate gamma random vectors are strongly positive upper orthant dependent and then integrated on both sides of the corresponding inequality.…”

“….} when the covariance matrix only has nonnegative entries (without the nonsingularity assumption), using an Isserlis-Wick type formula due to Song & Lee [29,30]; see also Corollary 5 of Mamis [17] and Remark 2.3 of Edelmann et al [5].…”

“…Email addresses: christian.genest@mcgill.ca (Christian Genest), frederic.ouimet2@mcgill.ca (Frédéric Ouimet) In parallel to these developments, Edelmann et al [5] recently established the analog of inequality (1) (without the nonsingularity assumption) for all α 1 , . .…”

Using Sums-of-Squares to Prove Gaussian Product Inequalities

“…, U n ) is an n-dimensional standard Gaussian random vector. Recently, Edelmann et al[3] used a different method to extend[14, Lemma 2.3] to the multivariate gamma distribution. It is interesting to point out…”

An Opposite Gaussian Product Inequality

“…n ) is an n-dimensional standard Gaussian random vector. Additionally, Edelmann et al[4] used a different method to extend[16, Lemma 2.3] to the multivariate gamma distribution.…”