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### Spectra of Random Correlation Matrices

What is then the spectrum of the correlation matrix and how does this effect our estimation for correlation?

This has been the question at the end of my last post - the answer to this question is known for several cases due to the work of Marcenko, Pastur [V. A. Marcenko and L. A. Pastur, Math. USSR-Sb, 1, 457-483 (1967)] and several others (see for example [Z. Burda, A. G ̈orlich, A. Jarosz and J. Jurkiewicz, Physica A, 343, 295-310 (2004)]).

Considering an empirical correlation matrix E of N assets using T data points, both very large, with q = N/T finite and the "true" correlation matrix C=, defining the Wishart ensemble. In statistics, the Wishart distribution is a generalization to multiple dimensions of the chi-squared distribution. It is part of a family of probability distributions defined over symmetric, nonnegative-definite matrix-valued random variables (“random matrices”).

Then for any choice (of course this choice  needs to obey the properties a correlation matrix needs to fulfil) of C one can obtain the density of eigenvalues, at least numerically. In some cases the density of the eigenvalues can be calculated analytically, so for C=I.

But are the results correct in every case ?  - What will happen if we consider a matrix with one large eigenvalue, separated from the "sea", the rest of the eigenvalues form. It has been shown, that the statistics of this isolated eigenvalue is Gaussian. So the Marcenko-Pastur results describe only  continuous parts of the spectrum but do not apply to isolated eigenvalues.

Empirical eigenvalue density for 406 stocks from the S&P 500, and fit using the MP distribution. Note the presence of one large eigenvalue  corresponding to the market mode. (Picture taken from Financial Applications of Random Matrix Theory: Old Laces and New Pieces, Potters et al.)