On eigenvalues of a high-dimensional Kendall's rank correlation matrix with dependence

TL;DR

This paper studies the spectral distribution of high-dimensional Kendall's rank correlation matrices with dependent data, revealing new laws different from classical results and exploring their relation to covariance matrices.

Contribution

It introduces the first analysis of eigenvalues of rank correlation matrices under dependence, extending spectral theory beyond independent cases.

Findings

The limiting spectral distribution deviates from the Marčenko-Pastur law under dependence.

Results apply to multivariate normal distributions with general covariance.

Provides insights into the relationship between Kendall's rank correlation and sample covariance matrices.

Abstract

This paper investigates limiting spectral distribution of a high-dimensional Kendall's rank correlation matrix. The underlying population is allowed to have general dependence structure. The result no longer follows the generalized Mar\u{c}enko-Pastur law, which is brand new. It's the first result on rank correlation matrices with dependence. As applications, we study the Kendall's rank correlation matrix for multivariate normal distributions with a general covariance matrix. From these results, we further gain insights on Kendall's rank correlation matrix and its connections with the sample covariance/correlation matrix.