Spectral statistics of high dimensional sample covariance matrix with unbounded population spectral norm

TL;DR

This paper develops new central limit theorems for spectral statistics of high-dimensional sample covariance matrices with unbounded population spectral norms, revealing how divergence rates influence fluctuations.

Contribution

It introduces novel CLTs for spectral statistics under divergent spectral norm models, including the divergent spiked population case, with variable numbers of spikes.

Findings

Divergence of population spectral norm affects spectral statistic fluctuations.

Theorems cover fixed and growing numbers of spiked eigenvalues.

Fluctuation behavior depends on divergence rate.

Abstract

In this paper, we establish some new central limit theorems for certain spectral statistics of a high-dimensional sample covariance matrix under a divergent spectral norm population model. This model covers the divergent spiked population model as a special case. Meanwhile, the number of the spiked eigenvalues can either be fixed or grow to infinity. It is seen from our theorems that the divergence of population spectral norm affects the fluctuations of the linear spectral statistics in a fickle way, depending on the divergence rate.