Independence of linear spectral statistics and the point process at the edge of Wigner matrices

In the current paper we consider a Wigner matrix and consider an analytic function of polynomial growth on a set containing the support of the semicircular law in its interior. We prove that the linear spectral statistics corresponding to the function and the point process at the edge of the Wigner matrix are asymptotically independent when the entries of the Wigner matrix are sub-Gaussian. The main ingredient of the proof is based on a recent paper by Banerjee [6]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.