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

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.