On spectral properties of large dimensional correlation matrices and covariance matrices computed from elliptically distributed data

October, 2007
Report Number: 
740
Authors: 
Noureddine El Karoui
Abstract: 

We place ourselves in the setting of high-dimensional statistical inference, where the number of variables p in a dataset 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 under the setting that p/n tendsto rho in (0,infinity), for general population covariance.

We show that spectral properties for large dimensional correlation matrices are similar to those of large dimensional covariance matrices, for a large class of models studied in random matrix theory.

We also derive a Marcenko-Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from elliptically distributed data. The motivation for this study comes from the relevance of such distributional assumptions to problems in econometrics and portfolio optimization.

A mathematical theme of the paper is the important use we make of concentration inequalities.

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