Limiting spectral distributions of large consistent rank correlation matrices

TL;DR

This paper proves that large matrices built from certain rank correlation measures, like Hoeffding's D, have spectral distributions following the semicircle law, contrasting with previous results for Kendall's tau and Spearman's rho.

Contribution

It generalizes recent findings by establishing the semicircle law for a broad class of consistent rank correlations in high-dimensional settings.

Findings

Hoeffding's D matrices follow the semicircle law in high dimensions.

Kendall's tau and Spearman's rho matrices converge to the Marchenko-Pastur law.

The result extends previous work on Chatterjee's rank correlation.

Abstract

We study random matrices whose entries are obtained by applying consistent rank correlations, such as Hoeffding's $D$, pairwise to a high-dimensional random vector with mutually independent components. Prior work has shown that, in the proportional high-dimensional regime, the empirical spectral distributions of large Kendall's tau and Spearman's rho matrices converge weakly almost surely to the Marchenko--Pastur law. By contrast, we prove that for consistent rank correlations such as Hoeffding's $D$, the limiting spectral distribution is given by the semicircle law. Our result thus generalizes a recent work of Dong, Han, and Yao (2025), who considered the special case of Chatterjee's rank correlation and established the first semicircle law for a large correlation matrix in the proportional regime.