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. As the basic example, we consider the GUE matrices. Immediate applications include GOE and the ensemble of Gaussian anti-symmetric Hermitian matrices. The estimates we derive give asymptotically exact expressions for the first terms of 1/N-expansions of the moments and covariance terms. We apply our method to the ensemble of Gaussian Hermitian random band matrices whose elements are zero outside of the band of width b. The estimates we obtain show that the spectral norm of these matrices remains bounded in the limit of infinite N when b is much greater than log N to the power 3/2.