On Non-Gaussian Limiting Laws for the Certain Statistics of the Wigner Matrices

We continue investigations of our previous papers, in which there were proved central limit theorems (CLT) for linear eigenvalue statistics Tr f(M_n) and there were found the limiting probability laws for the normalised matrix elements of differential functions f(M_n) of nxn real symmetric Wigner matrices M_n. Here we consider another spectral characteristic of Wigner matrices, namely, Tr f(M_n)A_n, where {A_n, n=1,2,...} is a certain sequence of non-random matrices. We show first that if M_n belongs to the Gaussian Orthogonal Ensemble, then Tr f(M_n)A_n satisfies CLT. Then we consider Wigner matrices with i.i.d. entries possessing entire characteristic function and find the limiting probability law for Tr f(M_n)A_n, which in general is not Gaussian.