CLT for LSS of sample covariance matrices with unbounded dispersions

TL;DR

This paper establishes a central limit theorem for linear spectral statistics of high-dimensional sample covariance matrices without assuming bounded population covariances, accommodating various kernel functions and diverging spectral norms.

Contribution

It extends existing CLT results by removing boundedness assumptions and allowing diverse kernel functions and diverging spectral norms in high-dimensional settings.

Findings

CLT derived for LSS without bounded covariance assumption

Applicable to logarithmic and polynomial kernel functions

Asymptotic mean and covariance relate to spectral norm divergence

Abstract

Under the high-dimensional setting that data dimension and sample size tend to infinity proportionally, we derive the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix. Different from existing literature, our results do not require the assumption that the population covariance matrices are bounded. Moreover, many common kernel functions in the real data such as logarithmic functions and polynomial functions are allowed in this paper. In our model, the number of spiked eigenvalues can be fixed or tend to infinity. One salient feature of the asymptotic mean and covariance in our proposed central limit theorem is that it is related to the divergence order of the population spectral norm.