An explicit Wishart moment formula for the product of two disjoint principal minors

TL;DR

This paper derives an explicit formula for the expectation of the product of two disjoint principal minors of a Wishart matrix, addressing a longstanding problem and introducing new inequalities related to Wishart matrices.

Contribution

It provides the first explicit Wishart moment formula for disjoint principal minors and formulates a Wishart generalization of the Gaussian product inequality conjecture.

Findings

Explicit Wishart moment formula derived

A Wishart generalization of the Gaussian product inequality proved

Stronger quantitative bounds established for two minors

Abstract

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.