CLT for linear spectral statistics of hermitian wigner matrices with general moment conditions

Abstract

In this paper, we study the Hermitian Wigner matrix Wn = (xij)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 Wn. Our result extends the previous results on this topic to a more general case without the assumption Ex2ij = 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 Ex2ij 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 Ex2ij ∈ [−1, 1].