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| 三维高斯乘积不等式的新结果（IV） |
| The New Results of Three-Dimensional Gaussian Product Inequality (IV) |
| 投稿时间：2024-10-12 修订日期：2025-01-10 |
| DOI： |
| 中文关键词: 高斯乘积不等式、正态分布、相关系数 |
| 英文关键词: Gaussian product inequality, normal distribution, correlation functions |
| 基金项目:海南省自然科学基金（122MS056，124MS056） |
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| 摘要点击次数: 78 |
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| 中文摘要: |
| 设 X_1, X_2, X_3为中心化的高斯随机变量，其协方差矩阵的对角线元素均为1。本文研究了三维高斯乘积不等式猜想:对任意值中心高斯随机向量和正整数，有三维高斯乘积不等式成立。
当均为正偶数时，上述三维高斯乘积不等式已经得到了证明。本文重点考虑了至少有一个为奇数的情形，首先借助于相关系数、偏相关系数、反正弦函数得到上述不等式左侧的表达式，然后分类讨论得到左侧的极小值，进而得到了： a_1+a_2+a_3小于6 时，高斯乘积不等式是成立的，等号成立当且仅当 X_1, X_2, X_3相互独立，从而补充了现有文献中有关高斯乘积不等式的结果。 |
| 英文摘要: |
| Let X_1, X_2, X_3 be a centered Gaussian random vector with .This paper attempts to prove the following special Gaussian product inequality conjecture: for any real-valued center Gaussian random vector and positive integers , the following Gaussian inequality holds,
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When all of are positive even integer, the above inequality has been proved. This paper focuses on the case that more than one of the integers are positive odd numbers. Firstly, the product expectation in the above inequality is presented in terms of the correlation coefficients, partial correlation coefficients and the arcsine function. Secondly, the minimum of the product expectation is obtained by classification discussion. Lastly, it is pointed that for a_1+a_2+a_3 less than 6 , Gaussian inequality holds and the equality holds if and only if X_1, X_2, X_3 are mutually independent. These results supplement the Gaussian product inequality in the existing literature. |
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