A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

Abstract

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector $\mathbf{X}=(X_1,\dots ,X_d)$ of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on $\mathbf{X}$ is shown to be strictly weaker than the assumption that the density of the random vector $(|X_1|,\dots ,|X_d|)$ is multivariate totally positive of order $2$, abbreviated MTP${}_2$, for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

9 pages, 1 figure

References in corpus (3)

• Squared chaotic random variables: new moment inequalities with applications
• Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions
• Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities

Cited by in corpus (3)

• A short proof of the strong three dimensional Gaussian product inequality
• Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices
• An explicit Wishart moment formula for the product of two disjoint principal minors