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Proof of Gaussian moment product conjecture

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.