Eur. Phys. J. B 73, 13-22 (2010)
https://doi.org/10.1140/epjb/e2009-00360-7

Spectral densities of Wishart-Lévy free stable random matrices

Analytical results and Monte Carlo validation

¹ Department of Physics, University of Milan, via Giovanni Celoria 16, 20133 Milano, Italy
² Department of Chemistry and WZMW, Computer Simulation Group, Philipps-University Marburg, 35032 Marburg, Germany
³ Department of Advanced Sciences and Technology, Laboratory on Complex Systems, Amedeo Avogadro University of Eastern Piedmont, Viale Teresa Michel 11, 15121 Alessandria, Italy
⁴ Institute for Scientific Interchange, Complex Systems Lagrange Lab, Viale Settimio Severo 65, 10133 Torino, Italy

Corresponding author: guido.germano@staff.uni-marburg.de

Received: 9 March 2009
Revised: 22 July 2009
Published online: 27 October 2009

Abstract

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.