Estimates for moments of random matrices with Gaussian elements

We describe an elementary method to get non-asymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. We derive a system of recurrent relations for the moments and the covariance terms and develop a triangular scheme to prove the recurrent estimates. The estimates we obtain are asymptoti-cally exact in the sense that they give exact expressions for the first terms of 1/N-expansions of the moments and covariance terms. As the basic example, we consider the Gaussian Unitary Ensemble of random matrices (GUE). Immediate applications include Gaussian Ortho-gonal Ensemble and the ensemble of Gaussian anti-symmetric Hermitian matrices. Finally we apply our method to the ensemble of N×N Gaussian Hermitian random matrices H(N,b) whose elements are zero outside of the band of width b. The other elements are taken from GUE; the matrix obtained is normalized by b−1/2. We derive the estimates for the moment