Spectral densities of Wishart-Lévy free stable random matrices: Analytical results and Monte Carlo validation

Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Lévy stable distributions. Here the derivation of an analytical expression for the spectra of covariance matrices approximated by free Lévy stable random variables is reviewed comprehensively and validated by Monte Carlo simulation.