Before revision: I apply the method of planar diagrammatic expansion - introduced in a self-consistent way - to solve the problem of finding the mean spectral density of the Hermitian equal-time and non-Hermitian time-lagged cross-covariance estimators, for systems of Gaussian random variables with various underlying covariance functions, both writing the general equations and applying them to several toy models. The models aim at a more and more accurate description of complex financial systems - to which a lengthy introduction is given - albeit only within the Gaussian approximation .
After revision: I apply the method of planar diagrammatic expansion - introduced in a self-consistent way - to solve the problem of finding the mean spectral density of the Hermitian equal-time and non-Hermitian time-lagged cross-covariance estimators, firstly in the form of master equations for the most general multivariate Gaussian system, secondly for seven particular toy models of the true covariance function. For the simplest one of these models, the existing result is shown to be incorrect and the right one is presented, moreover its generalizations are accomplished to the exponentially-weighted moving average estimator as well as two non-Gaussian distributions, Student t and free Levy. The paper revolves around applications to financial complex systems, and the results constitute a sensitive probe of the true correlations present there .