海南师范大学学报（自然科学版）2025，Vol.38Issue(2)：134-143,10.DOI:10.12051/j.issn.1674-4942.2025.02.003
三维高斯乘积不等式的新结果(Ⅳ)
The New Results of Three-Dimensional Gaussian Product Inequality(Ⅳ)
摘要
Abstract
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(n∏j=1|Xj|αj)≥n∏j=1E(|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.关键词
高斯乘积不等式/正态分布/相关系数Key words
Gaussian product inequality/normal distribution/correlation coefficient分类
数理科学引用本文复制引用
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（自然科学版）,2025,38(2):134-143,10.基金项目
海南省自然科学基金项目(122MS056,124MS056) （122MS056,124MS056）
