从格拉姆矩阵不等式到多维协方差不等式

本文先证明格拉姆矩阵分块矩阵满足的行列式、迹不等式,它们可以看作经典柯西-施瓦茨不等式的多维推广.作为推论,得到分别以协方差矩阵的子矩阵的行列式和迹表示的协方差矩阵不等式,它们是一维协方差不等式的多维推广.

协方差不等式的证明和推论

利用构造二次函数的方法,证明了改进的协方差不等式,并在证明过程中得到3个推论,进一步探讨了随机变量具有线性关系时的若干结论.

协方差不等式的若干证明及其注记——兼论大学数学各学科教学的融合

介绍协方差不等式的十种证明并指出证明之间的一些联系.证明分别用到数学中的归一化、不相关化、构造辅助函数、投影、恒等式、求极值等常见思想和方法,还涉及商空间、内积空间等概念.这些证明充分展示了分析、代数、几何与概率论及数理统计之间的密切联系.由此强调习惯上分门别类进行的大学数学各学科的教学应当相互融合,以使学生感知到数学是一个有机整体.

利用Cauchy-Schwarz不等式估计回归系数

根据随机变量情形下的Cauchy-Schwarz不等式,推广得到关于二阶矩的推论:随机变量X、Y的方差均存在且不为零,如果随机变量X、Y依概率1具有线性关系P{Y=aX+b}=1,则a=E(X-EX)(Y-EY)/E(X-EX)2,b=EY-aEX,同时给出线性回归问题中回归系数的一种矩估计法。

一种新的弱相依系数和应用(英文)

本文在积分概率距离意义下提出了两个随机变量之间一种新的弱相依系数,并证明了此系数可获得协方差不等式和强大数定律,而且对于相关随机变量序列,我们还可以进一步研究矩不等式.

Some conditional results for conditionally strong mixing sequences of random variables

The relation between strong mixing and conditionally strong mixing is answered by examples,that is,the strong mixing property of random variables does not imply the conditionally strong mixing property,and the opposite implication is also not true.Some equivalent definitions and basic properties of conditional strong mixing random variables are derived,and several conditional covariance inequalities are obtained.By means of these properties and conditional covariance inequalities,a conditional central limit theorem stated in terms of conditional characteristic functions is established,which is a conditional version of the earlier result under non-conditional case.