On eigenvalues of a renormalized sample correlation matrix

TL;DR

This paper investigates the spectral properties of a renormalized sample correlation matrix in high dimensions, deriving asymptotic distributions and proposing an independence test effective across various dimensional regimes.

Contribution

It provides a unified framework for spectral analysis of renormalized correlation matrices and introduces a new independence test based on the CLT for linear spectral statistics.

Findings

Derived the limiting spectral distribution for the matrix.

Established a CLT for linear spectral statistics.

Proposed an effective independence test for high-dimensional data.

Abstract

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 $p/n\rightarrow c\in (0,\infty]$. 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.