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 .
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 .
A
- introduced in a self-consistent way -
A
Hermitian equal-time and
R
covariance estimator for a system of i.i.d.
cross-covariance estimators, for systems of
R
. 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
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