Before revision: I apply the method of planar diagrammatic expansion  to solve the problem of finding the mean spectral density of the  non-Hermitian time-lagged covariance estimator for a system of i.i.d. Gaussian random variables . I confirm the result in a much simpler way using a recent conjecture about non-Hermitian random matrix modelswith rotationally-symmetric spectra. I conjecture and test numerically a form of finite-size corrections to the mean spectral density featuring the complementary error function .
After revision: I apply the method of planar diagrammatic expansion to solve the problem of finding the mean spectral density of the non-Hermitian time-lagged covariance estimator for a system of i.i.d. 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 .