Asymptotic Normality of Chatterjee's Rank Correlation

We prove that a suitably de-biased version of Chatterjee's rank correlation based on i.i.d. copies of a random vector $(X,Y)$ is asymptotically normal whenever $Y$ is not almost surely constant. No further conditions on the joint distribution of $X$ and $Y$ are required. We establish several results which allow us to extend convergence of the empirical process from one function class to larger function classes. These results are of independent interest, and can be used to investigate $V$-statistics and $V$-processes -- or, closely related, $U$-statistics and $U$-processes -- with dependent sample data. As an example, we use these results to prove weak convergence of $V$- and $U$-processes based on strongly mixing data. This implies a new limit theorem for $V$- and $U$-statistics of strongly mixing data.