Math Conjectures

Gaussian Product Inequality Conjecture

OPENMajorConjectureProposed 2012 · Full conjecture

Canonical statement

For every integer n≥1n\ge1, every centered jointly Gaussian random vector X=(X1,…,Xn)X=(X_1,\ldots,X_n) with Var⁡(Xi)>0\operatorname{Var}(X_i)>0 for all ii, and every choice of real exponents α1,…,αn>0\alpha_1,\ldots,\alpha_n>0,
E ⁣[∏i=1n∣Xi∣αi]≥∏i=1nE ⁣[∣Xi∣αi]. \mathbf E\!\left[\prod_{i=1}^{n}|X_i|^{\alpha_i}\right] \ge \prod_{i=1}^{n}\mathbf E\!\left[|X_i|^{\alpha_i}\right].
View source LaTeX
For every integer \(n\ge1\), every centered jointly Gaussian random vector \(X=(X_1,\ldots,X_n)\) with \(\operatorname{Var}(X_i)>0\) for all \(i\), and every choice of real exponents \(\alpha_1,\ldots,\alpha_n>0\),
\[
\mathbf E\!\left[\prod_{i=1}^{n}|X_i|^{\alpha_i}\right]
\ge
\prod_{i=1}^{n}\mathbf E\!\left[|X_i|^{\alpha_i}\right].
\]
STAT-002:[LiWei2012GaussianProducts][KimKimKim2025GaussianProducts]
StatisticsProbability theoryAnalysis and functional analysis

Notes

The inequality is known in two dimensions and for several structured covariance and exponent regimes, including important integer-moment cases. It remains open for arbitrary dimension, arbitrary covariance matrix, and arbitrary positive real exponents.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.