Limit theorems of Chatterjee's rank correlation
This provides foundational statistical theory for rank-based correlation measures, addressing a long-awaited problem in nonparametric statistics.
The paper establishes the asymptotic normality of Chatterjee's rank correlation for non-independent random variables, showing it holds unless one variable is a measurable function of the other, with an asymptotic variance bounded by 36 and a consistent variance estimator.
Establishing the limiting distribution of Chatterjee's rank correlation for a general, possibly non-independent, pair of random variables has been eagerly awaited by 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, (b) the corresponding asymptotic variance is uniformly bounded by 36, and (c) a consistent variance estimator exists. 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 Hájek representation and Chatterjee's nearest-neighbor CLT.