Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond

TL;DR

This paper investigates the spectral properties of high-dimensional correlation and covariance matrices, demonstrating their similarities under certain conditions and deriving new spectral distribution equations for elliptical distributions, with applications in econometrics and finance.

Contribution

It extends random matrix theory to high-dimensional correlation matrices, deriving new spectral distribution equations for elliptical distributions and highlighting the role of concentration inequalities.

Findings

Spectral properties of correlation matrices resemble those of covariance matrices in high dimensions.

Derived Marenko–Pastur-type equations for elliptical distribution data.

Demonstrated relevance to econometrics, portfolio optimization, and robustness of classical results.

Abstract

We place ourselves in the setting of high-dimensional statistical inference, where the number of variables $p$ in a data set of interest is of the same order of magnitude as the number of observations $n$. More formally, we study the asymptotic properties of correlation and covariance matrices, in the setting where $p/n\to\rho\in(0,\infty),$ for general population covariance. We show that, for a large class of models studied in random matrix theory, spectral properties of large-dimensional correlation matrices are similar to those of large-dimensional covarance matrices. We also derive a Mar\u{c}enko--Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.