Computing the Moments of the Complex Gaussian: Full and Sparse Covariance Matrix

Given a multivariate complex centered Gaussian vector Z=(Z(1), ... , Z(p)) with non-singular covariance matrix Sigma, we derive sufficient conditions on the nullity of the complex moments and we give a closed-form expression for the non-null complex moments. We present conditions for the factorisation of the complex moments. Computational consequences of these results are discussed.