On asymptotic expansion and CLT of linear eigenvalue statistics for sample covariance matrices when $N/M\to 0$

Abstract

We study the renormalized real sample covariance matrix $H=X^{T}X/\sqrt{MN}-\sqrt{M/N}$ with $N/M\to 0$ as $N,M\to \infty$ 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.

24 pages

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Rigidity of Eigenvalues of Generalized Wigner Matrices