关于条件不相关与随机变量独立性等价性的注记
A note on the equivalence between the conditional uncorrelation and the independence of random variables
AI总结：
本文证明在任意分位数条件集下条件不相关可推出随机变量独立，并给出两个示例说明其应用。
摘要：
众所周知，随机变量的独立性意味着零相关，但反之不成立。即，不相关的随机变量未必独立。本文证明，若考虑相关系数的局部化版本，则该蕴含关系可以逆转。更具体地，我们证明若随机变量在任意分位数条件集下条件（局部）不相关，则它们独立。为简化起见，我们聚焦于绝对连续情形。此外，我们通过两个简单例子说明所述结果的潜在用途。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
