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 strictly weaker than the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive of order $2$, abbreviated $\mbox{MTP}_2$, 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.

翻译：高斯产品不平等的组合证明(GPI)是在以下假设下给出的:任意长度的高斯中央随机矢量的每个组成部分 $\boldsymbol{X} = (X_1,\ldots, X_d) = (X_1,\ldots, X_d) 可以是标准高斯随机矢量组件的线性组合,具有相同符号的系数。这个条件$\boldsymbol{X} $ 的条件被证明严格弱于以下假设,即随机矢量的密度 $({X_1},\ldots, {X_d}$ =(x_d}) = (X_d_d) = (x_1) =(x_1) $, 缩放 $\mbox{MTP}2$, GPI 已经知道对此持有。在此条件下, 论文强调了GPI 和某些伽马函数的单比之间的新联系。