On eigenvalues of a renormalized sample correlation matrix

This paper studies the asymptotic spectral properties of a renormalized sample correlation matrix, including the limiting spectral distribution, the properties of largest eigenvalues, and the central limit theorem for linear spectral statistics. All asymptotic results are derived under a unified framework where the dimension-to-sample size ratio $p\mathrm{/}n\to c\in (0,\mathrm{\infty }]$. Based on our CLT result, we propose an independence test statistic capable of operating effectively in both high and ultrahigh dimensional scenarios. Simulation experiments demonstrate the accuracy of theoretical results.