We consider an N by N real or complex generalized Wigner matrix HN​,
whose entries are independent centered random variables with uniformly bounded
moments. We assume that the variance profile, sij​:=E∣Hij​∣2,
satisfies ∑i=1N​sij​=1, for all 1≤j≤N and c−1≤Nsij​≤c for all 1≤i,j≤N with some constant c≥1. We
establish Gaussian fluctuations for the linear eigenvalue statistics of HN​
on global scales, as well as on all mesoscopic scales up to the spectral edges,
with the expectation and variance formulated in terms of the variance profile.
We subsequently obtain the universal mesoscopic central limit theorems for the
linear eigenvalue statistics inside the bulk and at the edges respectively.Comment: Shortened the statement with refined proof. Updated the references
and corrected some typo