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On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices

Christian Genest, Frédéric Ouimet, Donald Richards

TL;DR

This work extends the Gaussian Product Inequality to the setting of disjoint principal minors of Wishart random matrices by leveraging matrix-variate completely monotone and Bernstein function frameworks and a matrix-Laplace transform order. It proves GPI-type product inequalities for disjoint blocks, derives explicit bounds for inverse determinant powers with precise finiteness conditions, and shows that the eigenvalues of Wishart matrices satisfy a GPI through their MTP$_2$ density. An elliptically distributed extension is proposed, providing a natural research direction that unifies Gaussian and non-Gaussian elliptical laws. Collectively, these results broaden the applicability of GPI in random matrix theory and multivariate statistics, with potential implications for high-dimensional probability and related fields.

Abstract

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 (MTP${}_2$). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.

On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices

TL;DR

This work extends the Gaussian Product Inequality to the setting of disjoint principal minors of Wishart random matrices by leveraging matrix-variate completely monotone and Bernstein function frameworks and a matrix-Laplace transform order. It proves GPI-type product inequalities for disjoint blocks, derives explicit bounds for inverse determinant powers with precise finiteness conditions, and shows that the eigenvalues of Wishart matrices satisfy a GPI through their MTP density. An elliptically distributed extension is proposed, providing a natural research direction that unifies Gaussian and non-Gaussian elliptical laws. Collectively, these results broaden the applicability of GPI in random matrix theory and multivariate statistics, with potential implications for high-dimensional probability and related fields.

Abstract

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 (MTP). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.
Paper Structure (8 sections, 15 theorems, 115 equations)

This paper contains 8 sections, 15 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Main results
  4. Proofs
  5. Possible extension: GPI for elliptical laws
  6. Technical lemmas
  7. Conditions for the finiteness of certain expectations
  8. Evaluation of the integrals in Corollary \ref{['cor:GPI.negative.powers.determinants.Wishart']}

Key Result

Proposition 2.9

Let $g: \mathcal{S}_{++}^p \to [0,\infty)$ be a matrix-variate Bernstein function with triplet $(0_{p\times p},B,\mu)$, where $\mu$ is assumed to be a probability measure on $\mathcal{S}_{++}^p$. If the map $\phi : [0,\infty) \to [0,\infty)$ is continuous, completely monotone on $(0,\infty)$ and sat

Theorems & Definitions (46)

  • Conjecture 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 36 more