标题： 关于Wishart随机矩阵不相交主子矩阵的高斯乘积不等式猜想 显示英文标题

标题： On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices

摘要： 本文将与高斯乘积不等式（GPI）猜想相关的各种结果推广到Wishart随机矩阵的不交主子式的框架下。这包括完全单调函数和Wishart不交主子式的Bernstein函数的矩阵变量类别的乘积型不等式，分别针对这两种情况。特别是，这些乘积型不等式适用于逆行列式的幂。当存在正指数和负指数的混合时，还得到了这些不等式的定量版本。此外，通过Wishart随机矩阵的特征值分布具有二阶多元总正性（MTP${}_2$），证明了GPI的一个扩展形式成立。此外，从椭圆分布的角度研究GPI提出了一个新的、尚未探索的研究方向。 显示英文摘要

摘要： 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.