This paper extends various results related to the Gaussian product inequality (GPI) conjecture to the setting of disjoint principal minors of Wishart random matrices. This includes product-type inequalities for matrix-variate analogs of completely monotone functions and Bernstein functions of Wishart disjoint principal minors, respectively. In particular, the product-type inequalities apply to inverse determinant powers. Quantitative versions of the inequalities are also obtained when there is a mix of positive and negative exponents. Furthermore, an extended form of the GPI is shown to hold for the eigenvalues of Wishart random matrices by virtue of their law being multivariate totally positive of order~2 ($\mathrm{MTP}_2$). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.

翻译：本文将高斯乘积不等式（GPI）猜想的若干相关结论拓展至Wishart随机矩阵不相交主子式的情形。具体而言，分别建立了完全单调函数与Bernstein函数的矩阵变量模拟关于Wishart不相交主子式的乘积型不等式。特别地，这些乘积型不等式适用于逆行列式幂次。当正负指数混合存在时，还得到了不等式的定量版本。此外，借助Wishart随机矩阵特征值分布具有~2阶多元全正性（$\mathrm{MTP}_2$）这一性质，证明了GPI的推广形式对该矩阵特征值成立。本文还提出了一条尚未探索的研究路径，即从椭圆分布视角研究GPI问题。