In this paper, we prove that the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.
"CLT for Linear Spectral Statistics of Wigner matrices." Electron. J. Probab. 14 2391 - 2417, 2009. https://doi.org/10.1214/EJP.v14-705