On asymptotic expansion and CLT of linear eigenvalue statistics for sample covariance matrices when $N/M\rightarrow0$

TL;DR

This paper investigates the asymptotic behavior and CLT of linear eigenvalue statistics for a specific sample covariance matrix model when the dimension ratio tends to zero, revealing Gaussian fluctuations with Wigner-like variance.

Contribution

It provides the first asymptotic expansion for the expectation of the Stieltjes transform and establishes a CLT for linear eigenvalue statistics in the regime where N/M approaches zero.

Findings

Asymptotic expansion of the expected Stieltjes transform is derived.

A CLT for linear eigenvalue statistics is proved.

Limiting variance matches that of Gaussian Wigner matrices.

Abstract

We study the renormalized real sample covariance matrix $H=X^TX/\sqrt{MN}-\sqrt{M/N}$ with $N/M\rightarrow0$ as $N, M\rightarrow \infty$ in this paper. And we always assume $M=M(N)$. Here $X=[X_{jk}]_{M\times N}$ is an $M\times N$ real random matrix with i.i.d entries, and we assume $\mathbb{E}|X_{11}|^{5+\delta}<\infty$ with some small positive $\delta$. The Stieltjes transform $m_N(z)=N^{-1}Tr(H-z)^{-1}$ and the linear eigenvalue statistics of $H$ are considered. We mainly focus on the asymptotic expansion of $\mathbb{E}\{m_N(z)\}$ in this paper. Then for some fine test function, a central limit theorem for the linear eigenvalue statistics of $H$ is established. We show that the variance of the limiting normal distribution coincides with the case of a real Wigner matrix with Gaussian entries.