Proof of Gaussian moment product conjecture

TL;DR

This paper proves the Gaussian moment product conjecture for any covariance matrix, establishing a fundamental inequality for moments of Gaussian vectors with broad implications.

Contribution

It provides a novel, elementary proof of the Gaussian moment product conjecture applicable to all covariance matrices.

Findings

Proved the Gaussian moment product conjecture for all Gaussian vectors.

Confirmed the conjecture's special case implies the real linear polarization constant.

Introduced an elementary proof technique based on varying variance of one component.

Abstract

For an $n$-dimensional real-valued centered Gaussian random vector $(X_1,\ldots,X_n)$ with any covariance matrix, the following moment product conjecture is proved in this paper \[ \mathbb{E}\prod_{j=1}^nX_j^{2m_j}\geq \prod_{j=1}^n\mathbb{E}X_j^{2m_j}, \] where $m_j\geq1,1\leq j\leq n,$ are any positive integers. Among other important applications, a special case of this conjecture (with $m_j=m,1\leq j\leq n$) would give an affirmative answer to another open problem: real linear polarization constant. The proof is based on a very elegant and elementary approach in which only one component $X_j$ of the random vector is chosen with varying variance.