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 (MTP${}_2$). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.

翻译：本文将与高斯乘积不等式(GPI)猜想相关的多种结果推广至Wishart随机矩阵不相交主子式的情形。这包括分别针对Wishart不相交主子式的完全单调函数矩阵变量类比及Bernstein函数建立的乘积型不等式。特别地，该乘积型不等式适用于逆行列式幂次。当指数存在正负混合时，我们还获得了不等式的定量版本。此外，基于Wishart随机矩阵律具有二阶多元完全正性(MTP${}_2$)的特性，证明了GPI的扩展形式对其特征值成立。本文提出了从椭圆分布角度研究GPI的新探索路径。