This paper provides the first explicit formula for the expectation of the product of two disjoint principal minors of a Wishart random matrix, solving a part of a broader problem put forth by Samuel S. Wilks in 1934 in the Annals of Mathematics. The proof makes crucial use of hypergeometric functions of matrix argument and their Laplace transforms. Additionally, a Wishart generalization of the Gaussian product inequality conjecture is formulated and a stronger quantitative version is proved to hold in the case of two minors.

翻译：本文首次给出了Wishart随机矩阵两个不相交主子式乘积期望的显式公式，部分解决了Samuel S. Wilks于1934年在《数学年刊》中提出的一个更广泛问题。证明过程关键性地运用了矩阵参数的超几何函数及其拉普拉斯变换。此外，本文提出了高斯乘积不等式猜想的Wishart推广形式，并证明了在两个主子式情形下成立的一个更强定量版本。