论文标题
关于条件不相关与随机变量独立性之间的等效性的注释
A note on the equivalence between the conditional uncorrelation and the independence of random variables
论文作者
论文摘要
众所周知，尽管随机变量的独立性意味着零相关性，但相反的情况并非如此。也就是说，不相关的随机变量不一定是独立的。在本说明中，我们表明，如果我们考虑相关系数的本地化版本，则可能会反转。更具体地说，我们表明，如果随机变量有条件地（本地）对于任何分位数调节集都不相关，则它们是独立的。为简单起见，我们专注于绝对连续的情况。另外，我们使用两个简单的示例说明了所述结果的潜在实用性。
It is well known that while the independence of random variables implies zero correlation, the opposite is not true. Namely, uncorrelated random variables are not necessarily independent. In this note we show that the implication could be reversed if we consider the localised version of the correlation coefficient. More specifically, we show that if random variables are conditionally (locally) uncorrelated for any quantile conditioning sets, then they are independent. For simplicity, we focus on the absolutely continuous case. Also, we illustrate potential usefulness of the stated result using two simple examples.
