Fluctuations of Matrix Elements of Regular Functions of Wigner and Sample Covariance Random Matrices

We consider nxn real symmetric and hermitian Wigner random matrices M with independent (modulo symmetry condition) entries, the (null) sample covariance matrices M=X*X with independent entries of mxn matrix X, and functions of these matrices f(M). Assuming that the 4rd moments of entries of initial random matrices are uniformly bounded, and the test functions f are smooth enough, we prove that as n tends to infinity, centered and multiplied by square root of n entries of f(M) satisfy the Central Limit Theorem with the same limiting variance as for matrices with Gaussian entries.