In this short note, we find an equivalent combinatorial condition only involving finite sums under which a centered Gaussian random vector with multinomial covariance matrix satisfies the Gaussian product inequality (GPI) conjecture. These covariance matrices are relevant since their off-diagonal elements are negative, which is the hardest case to cover for the GPI conjecture, as mentioned by [Russell, O., \& Sun, W. 2022. Some new {G}aussian product inequalities. {\em J. Math. Anal. Appl.}, {\bf 515}(2), Paper No. 126439, 21 pp.].

翻译：在这篇短文中，我们找到了一个仅涉及有限和的等价组合条件，在该条件下，具有多项式协方差矩阵的中心化高斯随机向量满足高斯乘积不等式（GPI）猜想。这些协方差矩阵的相关性在于其非对角元素为负，这是GPI猜想中最难涵盖的情况，正如[Russell, O., & Sun, W. 2022. Some new Gaussian product inequalities. J. Math. Anal. Appl., 515(2), Paper No. 126439, 21 pp.]所述。