A combinatorial proof of the Gaussian product inequality in 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 $\boldsymbol{X} = (X_1, \ldots, X_d)$ 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 $\boldsymbol{X}$ is shown to be equivalent (in a non-trivial way) to the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive density of order $2$ ($\mbox{MTP}_2$), for which the GPI is already known to hold. In addition to giving a new characterization of the $\mbox{MTP}_2$ class for nonsingular centered Gaussian random vectors, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.