Establishing limiting distributions of Chatterjee's rank correlation for a general, possibly non-independent, pair of random variables has been eagerly awaited to many. This paper shows that (a) Chatterjee's rank correlation is asymptotically normal as long as one variable is not a measurable function of the other, and (b) the corresponding asymptotic variance is uniformly bounded by 36. Similar results also hold for Azadkia-Chatterjee's graph-based correlation coefficient, a multivariate analogue of Chatterjee's original proposal. The proof is given by appealing to H\'ajek representation and Chatterjee's nearest-neighbor CLT.

翻译：许多人急切地等待着Chatterjee的一对可能非独立的普通随机变量建立限制等级相关性分布。 本文显示 (a) 只要一个变量不是另一个变量的可测量功能,Chatterjee的等级相关性就无异于正常, 以及 (b) 相应的非随机差异被36个一致地捆绑在一起。 Azadkia-Chatterjee的基于图表的关联系数,即Chatterjee最初提案的多变式类似系数, 也是同样的结果。 证据来自H\'ajek 代表和 Chatterjee的近邻CLT。