This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.
翻译:本说明报告了与Gaussian产品不平等(GPI)关于联合分配Wishart矩阵痕迹的假设有关的部分结果,特别是Wei(2014)和Liu等人(2015年)与GPI有关的若干结果以两种方式得到扩展:用更一般的功能类别取代权力功能,用更一般的跟踪Wishart分布假设取代通常的Gaussian和多变伽马分布假设,这些结论表明,GPI的Kronecker产品形式持有任何Wishart分布的对角区块。