On continuity of Chatterjee's rank correlation and related dependence measures

Abstract

While measures of concordance -- such as Spearman's rho, Kendall's tau, and Blomqvist's beta -- are continuous with respect to weak convergence, Chatterjee's rank correlation xi recently introduced in Azadkia and Chatterjee (2021) does not share this property, causing drawbacks in statistical inference as pointed out in B\"ucher and Dette (2025). As we study in this paper, xi is instead weakly continuous with respect to conditionally independent copies -- the Markov products. To establish weak continuity of Markov products, we provide several sufficient conditions, including copula-based criteria and conditions relying on the concept of conditional weak convergence in Sweeting (1989). As a consequence, we also obtain continuity results for xi and related dependence measures and verify their continuity in the parameters of standard models such as multivariate elliptical and l1-norm symmetric distributions.