Spectral analysis of large dimensional Chatterjee's rank correlation matrix

Abstract: This paper studies the spectral behavior of large dimensional Chatterjee's rank correlation matrix when observations are independent draws from a high-dimensional random vector with independent continuous components. We show that the empirical spectral distribution of its symmetrized version converges to the semicircle law, and thus providing the first example of a large correlation matrix deviating from the Marchenko-Pastur law that governs those of Pearson, Kendall, and Spearman. We further establish central limit theorems for linear spectral statistics, which in turn enable the development of Chatterjee's rank correlation-based tests of complete independence among the components.