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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 sameorder of magnitude as the number of observations $n$. More formally, we studythe asymptotic properties of correlation and covariance matrices, in thesetting where $p/n\to\rho\in(0,\infty),$ for general population covariance. Weshow that, for a large class of models studied in random matrix theory,spectral properties of large-dimensional correlation matrices are similar tothose of large-dimensional covarance matrices. We also derive aMar\u{c}enko--Pastur-type system of equations for the limiting spectraldistribution of covariance matrices computed from data with ellipticaldistributions and generalizations of this family. The motivation for this studycomes partly from the possible relevance of such distributional assumptions toproblems in econometrics and portfolio optimization, as well as robustnessquestions for certain classical random matrix results. A mathematical theme ofthe paper is the important use we make of concentration inequalities.

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