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 [5] does not share this property, causing drawbacks in statistical inference as pointed out in B\"ucher and Dette [7]. 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 [36]. 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.