Limit theorems of Chatterjee's rank correlation
Abstract
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 H\'ajek representation and Chatterjee's nearest-neighbor CLT.
Cite
@article{arxiv.2204.08031,
title = {Limit theorems of Chatterjee's rank correlation},
author = {Zhexiao Lin and Fang Han},
journal= {arXiv preprint arXiv:2204.08031},
year = {2025}
}
Comments
Multiple minor improvements were made in this version, including (1) a proof of the existence of the limiting variance, (2) some numeric studies, and (3) an analysis of the Sobol' indices