Mar\v{c}enko-Pastur Law for Kendall's Tau

TL;DR

This paper proves that the spectral distribution of Kendall's Tau correlation matrix converges to the Marčenko-Pastur law for i.i.d. continuous data, marking a novel result for multivariate U-statistics.

Contribution

It establishes the first theoretical result on the empirical spectral distribution of a multivariate U-statistic, specifically Kendall's Tau.

Findings

Kendall's Tau matrix converges to Marčenko-Pastur law

First spectral distribution result for multivariate U-statistics

Applicable under i.i.d. continuous data assumptions

Abstract

We prove that Kendall's Rank correlation matrix converges to the Mar\v{c}enko-Pastur law, under the assumption that the observations are i.i.d random vectors $X_1$, $\dots$, $X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate $U$-statistic.