| 马 丽，陈蓬颖，韩新方*.三维高斯乘积不等式的新结果（IV）[J].海南师范大学学报自科版,2025,38(2):134-143 |
| 三维高斯乘积不等式的新结果（IV） |
| The New Results of Three-Dimensional Gaussian Product Inequality (IV) |
| |
| DOI：10.12051/j.issn.1674-4942.2025.02.003 |
| 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 coefficient |
| 基金项目:海南省自然科学基金项目（122MS056，124MS056） |
|
| 摘要点击次数: 825 |
| 全文下载次数: 533 |
| 中文摘要: |
| 设(X1，X2，X3 )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R3值中心高斯随机向量(X1，X2，X3 )和正整数α1、α2、α3，有
E(∏
j = 1
n
| Xj |
αj
) ≥ ∏
j = 1
n
E(| Xj |
αj
)。当α1、α2、α3均为正偶数时，上述不等式已经得到了证明。本文重点考
虑了α1、α2、α3至少有1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α1 + α2 + α3 ≤ 6时高斯乘积不
等式是成立的，等号成立当且仅当X1、X2、X3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 (X1，X2，X3 ) be a centered Gaussian random vector with D(Xi ) = 1，i = 1，2，3. This paper attempts to
prove the following special Gaussian product inequality conjecture: for any real-valued center Gaussian random vector
(X1，X2，X3 ) and positive integers α1，α2，α3, the following Gaussian inequality holds, E(∏
j = 1
n
| Xj |
αj
) ≥ ∏
j = 1
n
E(| Xj |
αj
).
When all of α1，α2，α3 are positive even integer, the above inequality has been proved. This paper focuses on the case that
more than one of the integers α1，α2，α3 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 α1 + α2 + α3 ≤ 6,
Gaussian inequality holds and the equality holds if and only if X1，X2，X3 are mutually independent. These results supple⁃
ment the Gaussian product inequality in the existing literature. |
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