Limit theorems of Chatterjee's rank correlation

04/17/2022

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by   Zhexiao Lin, et al.

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Establishing the limiting distribution 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 Hájek representation and Chatterjee's nearest-neighbor CLT.