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

This paper is a continuation of our papers [12–14] in which the limiting laws of fluctuations were found for the linear eigenvalue statistics $\mathrm{Tr}\,\varphi (M^{(n)})$ and for the normalized matrix elements $\sqrt{n}\varphi_{jj}(M^{(n)})$ of differentiable functions of real symmetric Wigner matrices $M^{(n)}$ as $n\rightarrow\infty$. Here we consider another spectral characteristic of Wigner matrices, $\xi^{A} _{n}[\varphi ]=\mathrm{Tr}\,\varphi (M^{(n)})A^{(n)}$, where $\{A^{(n)}\}_{n=1}^\infty$ is a certain sequence of non-random matrices. We show first that if $M^{(n)}$ belongs to the Gaussian Orthogonal Ensemble, then $\xi^{A} _{n}[\varphi ]$ satisfies the Central Limit Theorem. Then we consider Wigner matrices with i.i.d. entries possessing the entire characteristic function and find the limiting probability law for $\xi^{A} _{n}[\varphi ]$, which in general is not Gaussian.