标题： 关于Wishart随机矩阵不相交主子式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扩展到Wishart随机矩阵的不相交主子式的设置中。这包括矩阵变量的完全单调函数和Bernstein函数的不相交主子式的乘积型不等式。特别是，这些乘积型不等式适用于逆行列式幂。当存在正负指数混合时，也获得了不等式的定量版本。此外，通过其分布为二阶多变量完全正相关的性质，证明了GPI的一种扩展形式适用于Wishart随机矩阵的特征值（$\mathrm{MTP}_2$）。还提出了一种新的、未被探索的研究途径，从椭圆分布的角度研究GPI。 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 ($\mathrm{MTP}_2$). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.