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The random matrix theory method of planar Gaussian diagrammatic expansion isapplied to find the mean spectral density of the Hermitian equal-time andnon-Hermitian time-lagged cross-covariance estimators, firstly in the form ofmaster equations for the most general multivariate Gaussian system, secondlyfor seven particular toy models of the true covariance function. For thesimplest one of these models, the existing result is shown to be incorrect andthe right one is presented, moreover its generalizations are accomplished tothe exponentially-weighted moving average estimator as well as two non-Gaussiandistributions, Student t and free Levy. The paper revolves around applicationsto financial complex systems, and the results constitute a sensitive probe ofthe true correlations present there.

Hermitian and non-Hermitian covariance estimators for multivariate Gaussian and non-Gaussian assets from random matrix theory