A non-Gaussian limit for linear eigenvalue statistics of Hankel matrices

This paper focuses on linear eigenvalue statistics of Hankel matrices with independent entries. Using the convergence of moments we show that the linear eigenvalue statistics of Hankel matrices for odd degree monomials with degree greater than or equal to three does not converge in distribution to a Gaussian random variable. This result is a departure from the known results by Liu, Sun and Wang [Fluctuations of eigenvalues for random Toeplitz and related matrices, Electron. J. Probab. 17(95) (2012) 22, MR 2994843], Kumar and Maurya [Asymptotic behaviour of linear eigenvalue statistics of Hankel matrices, Statist. Probab. Lett. 181 (2022) 109273, MR 4334692], of linear eigenvalue statistics of Hankel matrices for even degree monomial test functions, where the limits were Gaussian random variables.