Results related to the Gaussian product inequality conjecture for determinants of diagonal blocks of Wishart random matrices

In this paper, various results related to the Gaussian product inequality (GPI) conjecture, from Wei (2014), Genest & Ouimet (2023) and Zhou et al. (2023), are generalized to the setting of determinants of diagonal blocks of Wishart random matrices. An extended form of the GPI is also shown to hold for the eigenvalues of Wishart random matrices by virtue of their law being multivariate totally positive of order 2 (MTP 2). As final thoughts, a new unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.