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

TL;DR

This paper applies random matrix theory to derive spectral densities of Hermitian and non-Hermitian covariance estimators for multivariate assets, including Gaussian and non-Gaussian models, with applications to financial systems.

Contribution

It provides new analytical formulas for spectral densities of covariance estimators, correcting previous results and extending to non-Gaussian distributions and weighted estimators.

Findings

Corrected the spectral density for the simplest model.

Extended analysis to Student t and free Levy distributions.

Applied results to financial correlation analysis.

Abstract

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