发表机构
Université du Québec à Trois-Rivières（魁北克大学特罗伊斯里维耶斯分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了强高斯乘积不等式猜想，即中心化高斯向量绝对值的任意正幂次乘积期望不小于各分量期望之积，并由此解决了实线性极化常数猜想，推导出多项相关不等式与刻画。
AI 中文摘要
设 $\boldsymbol{X} = (X_1,\ldots,X_n)$ 为中心化高斯向量，不一定非退化。本文证明，对于所有 $\alpha_1,\ldots,\alpha_n > 0$，有 \\[ \mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right]. \\] 当所有边际方差为正时，等号成立当且仅当各坐标相互独立。这解决了 Frenkel 提出已有 18 年之久的高斯乘积不等式（GPI）猜想，并且实际上也解决了其后来加强到任意正指数的版本。通过 Frenkel 的 hafnian 表述，该结果还为已有 28 年历史的实线性极化常数猜想提供了简短证明，该猜想今年已作为强极化不等式的推论被解决。主要结果导出了精确的实线性极化常数、实线性泛函的尖锐加权乘积不等式、球面矩界、正半定矩阵的 hafnian 不等式及其等号情形的完整刻画、对数方差和协方差不等式、Rényi 总相关证书，以及先前以正指数 GPI 为条件的混合符号高斯矩不等式的无条件版本。
英文摘要
Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every $α_1,\ldots,α_n > 0$, \[ \mathsf{E}\left[\prod_{i=1}^n |X_i|^{α_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{α_i}\right]. \] When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a Rényi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.
Comments17 pages, 0 figures
