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

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X = (X-1, ..., 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 X is shown to be strictly weaker than the assumption that the density of the random vector (vertical bar X-1 vertical bar, ..., vertical bar X-d vertical bar) is multivariate totally positive of order 2, abbreviated MTP2, 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.

Automated 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.