In this paper, we find an equivalent combinatorial condition only involving finite sums (which is appealing from a numerical point of view) under which the centered Gaussian random vector with multinomial covariance, $(X_1,X_2,\dots,X_d) \sim \mathcal{N}_d(\boldsymbol{0}_d, \mathrm{diag}(\boldsymbol{p}) - \boldsymbol{p} \boldsymbol{p}^{\top})$, satisfies the Gaussian product inequality (GPI), namely $$\mathbb{E}\left[\prod_{i=1}^d X_i^{2m}\right] \geq \prod_{i=1}^d \mathbb{E}\left[X_i^{2m}\right], \quad m\in \mathbb{N}.$$ 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 & Sun (2022). Numerical computations provide evidence for the validity of the combinatorial condition.
翻译：在本文中,我们发现一个等效的组合条件仅涉及一定数量(从数字角度来说具有吸引力),根据这个条件,核心高斯随机矢量与多数值共差,$(X_1,X_2,\d,X_d)\mathcal{N ⁇ d(Boldsymbol{0 ⁇ d,\mathrm{diag}(Boldsymbol{p}) -\boldsymbol{p}\boldsymbol{p}}}(Boldsymbol{p ⁇ to})$,满足高斯产品不平等(GPI),即$\mathb{left{E ⁇ left[\p}[\prod ⁇ i=1 ⁇ d X_i%2m ⁇ right]\geq\ prod\ prod ⁇ i=1 ⁇ d\d\\\\\\\\\\\\\\\\\\\\\\\\\\\\\right}{(Eleft[X_rforst[x_i_i_i\i\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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