Some new results on Gaussian product inequalities

Article Mathematics, Applied

Quantitative versions of the two-dimensional Gaussian product inequalities

Ze-Chun Hu et al.

Summary: In this study, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. By employing the hypergeometric functions and the generalized hypergeometric functions, we prove that for any nondegenerate and centered two-dimensional Gaussian random vector (X1,X2) with a correlation coefficient rho, the inequality E[|X1|alpha1|X2|alpha2]-E[|X1|alpha1]E[|X2|alpha2]>= f(alpha1,alpha2,rho)>= 0 holds when alpha1 and alpha2 are either in (-1,0) or (0,infinity), and provide the explicit expression of the function f(alpha1,alpha2,rho). Furthermore, when -1<alpha1<0 and alpha2>0, we also prove the reverse Gaussian product inequality.

JOURNAL OF INEQUALITIES AND APPLICATIONS (2023)

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Article Statistics & Probability

Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions

Dominic Edelmann et al.

Summary: The Gaussian product inequality is an important conjecture about the moments of Gaussian random vectors. Although a full proof of the inequality has not been achieved, partial results have been obtained in recent decades. In this study, the authors provide further results on the problem, including a strong version of the inequality for multivariate gamma distributions with nonnegative correlations. The authors also show that the inequality holds for random vectors with positive components and nonnegative exponents when the underlying vector is positively upper orthant dependent.

STATISTICS & PROBABILITY LETTERS (2023)

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Article Mathematics, Applied

Moment ratio inequality of bivariate Gaussian distribution and three-dimensional Gaussian product inequality

Oliver Russell et al.

Summary: We prove the three-dimensional Gaussian product inequality (GPI) E[X12X22m2X2m3 3] >= E[X12]E[X22m2]E[X2m3 3] for any centered Gaussian random vector (X1, X2, X3) and m2, m3 is an element of N. We discover a novel inequality for the moment ratio |E[X 2m2+1X2m3+1 2 3]| E[X22m2X2m3 3], which implies the 3D-GPI. The interplay between computing and hard analysis plays a crucial role in the proofs.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2023)

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Article Mathematics, Applied

A SHORT PROOF OF A STRONG FORM OF THE THREE DIMENSIONAL GAUSSIAN PRODUCT INEQUALITY

Ronan Herry et al.

Summary: We have proven a strong form of the Gaussian product conjecture in three dimensions. Our purely analytical proof simplifies previous proofs that relied on combinatorial methods or computer-assisted methods, and it allows us to solve the case of any triple of even positive integers that has remained unsolved until now.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2023)

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Article Statistics & Probability

An opposite Gaussian product inequality

Oliver Russell et al.

Summary: In this note, we prove a novel opposite Gaussian product inequality for specific bivariate Gaussian random variables. This finding completes the theory of bivariate Gaussian product relations.

STATISTICS & PROBABILITY LETTERS (2022)

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Article Statistics & Probability

A combinatorial proof of the Gaussian product inequality beyond the MTP2 case

Christian Genest et al.

Summary: This article provides a combinatorial proof of the Gaussian product inequality (GPI) by utilizing the characteristics of centered Gaussian random vectors and the monotonicity of a certain ratio of gamma functions.

DEPENDENCE MODELING (2022)

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Article Mathematics, Applied