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Asymptotic Normality of Chatterjee's Rank Correlation

Marius Kroll

TL;DR

The paper addresses the limiting behavior of Chatterjee's rank correlation by proving asymptotic normality for a de-biased estimator under a minimal assumption that $Y$ is not almost surely constant. It develops an empirical-process framework that transfers convergence from indicator-function classes to broader function classes and extends to $V$- and $U$-processes for dependent data, including strongly mixing sequences. It shows degeneracy only when $Y$ is a measurable function of $X$, and provides sufficient bias-control conditions ensuring $\sqrt{n}$-consistency. The results yield practical limit theorems for Chatterjee’s statistic and related dependence measures in non-i.i.d. settings, with Kendall’s tau offered as a concrete illustration of the method’s reach and robustness.

Abstract

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.

Asymptotic Normality of Chatterjee's Rank Correlation

TL;DR

The paper addresses the limiting behavior of Chatterjee's rank correlation by proving asymptotic normality for a de-biased estimator under a minimal assumption that is not almost surely constant. It develops an empirical-process framework that transfers convergence from indicator-function classes to broader function classes and extends to - and -processes for dependent data, including strongly mixing sequences. It shows degeneracy only when is a measurable function of , and provides sufficient bias-control conditions ensuring -consistency. The results yield practical limit theorems for Chatterjee’s statistic and related dependence measures in non-i.i.d. settings, with Kendall’s tau offered as a concrete illustration of the method’s reach and robustness.

Abstract

We prove that a suitably de-biased version of Chatterjee's rank correlation based on i.i.d. copies of a random vector is asymptotically normal whenever is not almost surely constant. No further conditions on the joint distribution of and 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 -statistics and -processes -- or, closely related, -statistics and -processes -- with dependent sample data. As an example, we use these results to prove weak convergence of - and -processes based on strongly mixing data. This implies a new limit theorem for - and -statistics of strongly mixing data.
Paper Structure (12 sections, 26 theorems, 252 equations)

This paper contains 12 sections, 26 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Chatterjee's Rank Correlation and Empirical Processes
  4. Some Notes on the Bias
  5. V- and U-Processes of Strongly Mixing Data
  6. Proofs of Empirical Process Results
  7. Convergence Indexed in Indicator Functions
  8. Extension to Functions of Bounded Variation
  9. Extension to Product Measure Processes
  10. Proofs for Chatterjee's Rank Correlation
  11. Proofs for Mixing Sequences
  12. Functions of Bounded Variation

Key Result

Theorem 2.1

If $Y$ is not almost surely constant, then there exists a sequence of random variables $\delta_n$, $n \in \mathbb{N}$, such that $\delta_n \to 0$ almost surely, and for some $\sigma^2 \geq 0$ depending on the joint distribution of $X$ and $Y$. It holds that $\sigma^2 = 0$ if $Y = f(X)$ almost surely for some measurable $f$. The rate of convergence of $\delta_n$ is discussed in Section subsec:bias

Theorems & Definitions (50)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Corollary 2.1
  • Lemma 3.1
  • proof
  • ...and 40 more