CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions

Abstract

In this paper, we study the Hermitian Wigner matrix $W_n=(x_{ij})_{1\le i,j\le n}$ with independent (up to symmetry) mean zero variance one entries. Under some Lindeberg type condition on the fourth moments of the entries, we establish a central limit theorem for the linear eigenvalue statistics of $W_n$. Our result extends the previous results on this topic to a more general case without the assumption ${\bf E}\,x_{ij}^2=0$ for $1\le i<j\le n$. Instead, we only assume that the real part and imaginary part of the upper-diagonal entry are uncorrelated. More precisely, we require ${\bf E}\,x_{ij}^2$ to be real and homogeneous for all $1\le i<j\le n$. The limiting normal distribution of the central limit theorem is shown to depend on the parameter ${\bf E}\,x_{ij}^2\in[-1,1]$.