On Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices with Non-Identically Distributed Entries

In this note, we extend the results about the fluctuations of the matrix entries of regular functions of Wigner random matrices obtained in arXiv:1103.3731 [math.PR] to Wigner matrices with non-i.i.d. entries provided certain Lindeberg type conditions for the fourth moments of the off-diagonal entries and the second moments of the diagonal entries are satisfied. In addition, we relax our conditions on the test functions and require that for some $s>3\int (1+\mathrm{\mid }k\mathrm{\mid }{)}^{2s}\text{*}\mathrm{\mid }\widehat{f}(k){\mathrm{\mid }}^2\text{*}dk<\mathrm{\infty }\mathrm{.}$