Spectral properties of high dimensional rescaled sample correlation matrices

TL;DR

This paper investigates the spectral properties of high-dimensional rescaled sample correlation matrices, deriving their limiting spectral distribution and CLTs for linear spectral statistics under general conditions.

Contribution

It extends existing results by deriving spectral properties without requiring the matrix M to be identity, and establishes joint CLTs for multiple spectral statistics.

Findings

Derived the limiting spectral distribution of rescaled sample correlation matrices.

Established CLTs for linear spectral statistics under linear independent component and elliptical structures.

Provided an application demonstrating the CLT in practice.

Abstract

High-dimensional sample correlation matrices are a crucial class of random matrices in multivariate statistical analysis. The central limit theorem (CLT) provides a theoretical foundation for statistical inference. In this paper, assuming that the data dimension increases proportionally with the sample size, we derive the limiting spectral distribution of the matrix $\widehat{\mathbf{R}}_n\mathbf{M}$ and establish the CLTs for the linear spectral statistics (LSS) of $\widehat{\mathbf{R}}_n\mathbf{M}$ in two structures: linear independent component structure and elliptical structure. In contrast to existing literature, our proposed spectral properties do not require $\mathbf{M}$ to be an identity matrix. Moreover, we also derive the joint limiting distribution of LSSs of $\widehat{\mathbf{R}}_n \mathbf{M}_1,\ldots,\widehat{\mathbf{R}}_n \mathbf{M}_K$. As an illustration, an application is given for the CLT.