It is well known that the independent random variables $X$ and $Y$ are uncorrelated in the sense $E[XY]=E[X]\cdot E[Y]$ and that the implication may be reversed in very specific cases only. This paper proves that under general assumptions the conditional uncorrelation of random variables, where the conditioning takes place over the suitable class of test sets, is equivalent to the independence. It is also shown that the mutual independence of $X_1,\dots,X_n$ is equivalent to the fact that any conditional correlation matrix equals to the identity matrix.

翻译：众所周知，独立随机变量$X$和$Y$是不相关的，即满足$E[XY]=E[X]\cdot E[Y]$，且该蕴含关系仅在极特殊情况下可逆。本文证明，在一般性假设下，随机变量在适当测试集类上的条件不相关性等价于独立性。研究还表明，$X_1,\dots,X_n$的相互独立性等价于任意条件相关矩阵均等于单位矩阵这一事实。