CLT FOR LINEAR SPECTRAL STATISTICS OF HERMITIAN WIGNER MATRICES WITH GENERAL MOMENT CONDITIONS

In this paper, we study the Hermitian Wigner matrix W-n = (x(ij))(1 <= i, j <= 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 Ex(ij)(2) = 0 for 1 <= i < j <= n. Instead, we only assume that the real part and imaginary part of the upper-diagonal entry are uncorrelated. More precisely, we require Ex(ij)(2) to be real and homogeneous for all 1 <= i < j <= n. The limiting normal distribution of the central limit theorem is shown to depend on the parameter Ex(ij)(2) is an element of [-1, 1].