CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size

TL;DR

This paper establishes a CLT for linear spectral statistics of normalized sample covariance matrices when the dimension greatly exceeds the sample size, under certain moment conditions, and applies it to covariance matrix testing.

Contribution

It introduces a CLT for LSS of normalized covariance matrices in high-dimensional regimes where dimension exceeds sample size, extending previous results.

Findings

CLT holds for p/n→∞ under fourth moment conditions

Results enable testing if population covariance is identity

Applicable in high-dimensional statistical inference

Abstract

Let $\mathbf{A}=\frac{1}{\sqrt{np}}(\mathbf{X}^T\mathbf{X}-p\mathbf {I}_n)$ where $\mathbf{X}$ is a $p\times n$ matrix, consisting of independent and identically distributed (i.i.d.) real random variables $X_{ij}$ with mean zero and variance one. When $p/n\to\infty$, under fourth moment conditions a central limit theorem (CLT) for linear spectral statistics (LSS) of $\mathbf{A}$ defined by the eigenvalues is established. We also explore its applications in testing whether a population covariance matrix is an identity matrix.