Limiting Spectral Distribution of moderately large Kendall's correlation matrix and its application

TL;DR

This paper derives the limiting spectral distribution of Kendall's correlation matrices in moderate high-dimensional settings, accommodating heterogeneity and non-i.i.d. data, and applies it to dependence detection.

Contribution

It extends spectral distribution results to heterogenous, non-i.i.d. data and provides a new tool for dependence detection in high-dimensional data.

Findings

Spectral distribution converges to a model-dependent limit.

Heterogeneity affects the spectral distribution.

Ignoring heterogeneity can cause false dependence detection.

Abstract

We establish the limiting spectral distribution of Kendall's correlation matrices in the moderate high-dimensional regime where the dimension grows slower than the sample size. Our framework allows observations to be independent but not necessarily identically distributed, and accommodates both discrete and continuous data. Unlike existing results developed under i.i.d. observations, our approach remains valid under substantial distributional heterogeneity and also covers certain i.i.d. models beyond previously studied settings. Under mild symmetry and convergence conditions on some traces, we prove that the empirical spectral distribution of a properly centered and scaled Kendall's correlation matrix converges weakly almost surely to a deterministic, generally model-dependent limit. The analysis clarifies how distributional heterogeneity influences the limiting spectrum. As an application, we propose a graphical tool for detecting dependence among components in high-dimensional data and show that ignoring heterogeneity may lead to spurious detection of dependence.