  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.
