Product inequality for disjoint principal minors of Wishart matrices
For an integer $p \ge 1$, a real $\alpha > p - 1$ and a real symmetric positive definite $\Sigma \in \mathbb{R}^{p \times p}$, let $X \sim W_p(\alpha, \Sigma)$ be the random real symmetric positive definite $p \times p$ matrix with density proportional to $$(\det X)^{(\alpha - p - 1)/2} \exp\!\left(-\tfrac12 \operatorname{tr}(\Sigma^{-1} X)\right)$$ with respect to Lebesgue measure on its independent entries; $\alpha$ need not be an integer. For positive integers $p_1, \dots, p_d$ with $p_1 + \dots + p_d = p$, partition $X = (X_{ij})_{i,j=1}^d$ into blocks $X_{ij}$ of size $p_i \times p_j$. The inequality below is a Wishart analogue of the Gaussian product inequality, and all the expectations in it are finite. Genest, Ouimet and Richards proved it for $d = 2$, in a stronger quantitative form.
Problem. Is $$\mathbb{E}\left[\prod_{i=1}^d (\det X_{ii})^{\nu_i}\right] \ge \prod_{i=1}^d \mathbb{E}\left[(\det X_{ii})^{\nu_i}\right]$$ for all positive integers $d, p_1, \dots, p_d$, every real $\alpha > p - 1$, where $p = p_1 + \dots + p_d$, every real symmetric positive definite $\Sigma \in \mathbb{R}^{p \times p}$ and all real $\nu_1, \dots, \nu_d \ge 0$?
Source: Genest, Ouimet and Richards (2024), Conjecture 1.1.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- PR
- Keywords
- Gaussian product inequality, OpenProblemsInNLA, principal minors, Wishart matrices
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{genest2024,
author = {Genest, Christian and Ouimet, Fr{\'e}d{\'e}ric and Richards, Donald},
title = {On the {G}aussian product inequality conjecture for disjoint principal minors of {W}ishart random matrices},
journal = {Electron. J. Probab.},
volume = {29},
year = {2024},
pages = {Paper No. 166},
doi = {10.1214/24-EJP1222},
eprint = {2311.00202},
archivePrefix = {arXiv},
}
@article{genest2025,
author = {Genest, Christian and Ouimet, Fr{\'e}d{\'e}ric and Richards, Donald},
title = {An explicit {W}ishart moment formula for the product of two disjoint principal minors},
journal = {Proc. Amer. Math. Soc.},
volume = {153},
number = {3},
year = {2025},
pages = {1299--1311},
doi = {10.1090/proc/17077},
eprint = {2409.14512},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi30,
author = {Townsend, Alex},
title = {{MI-30}: Product inequality for disjoint principal minors of Wishart matrices},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-30/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00146, "Product inequality for disjoint principal minors of Wishart matrices", clopen.solutions.