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A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. (English) Zbl 1018.62042
Summary: Recently K. Johansson [Commun. Math. Phys. 209, No. 2, 437-476 (2000; Zbl 0969.15008)] and I. M. Johnstone [Ann. Stat. 29, No. 2, 295-327 (2001; Zbl 1016.62078)] proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix \(X^*X(X'X)\) converges to the Tracy-Widom law [C. A. Tracy and H. Widom, J. Stat. Phys. 92, No. 5-6, 809-835 (1998; Zbl 0942.60099)] as \(n,p\) (the dimensions of \(X)\) tend to \(\infty\) in some ratio \(n/p\to \gamma>0\). We extend these results in two directions.
First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy-Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices allows to extend the results by Johansson and Johnstone to the case of \(X\) with non-Gaussian entries, provided \(n-p=O(p^{1/3})\). We also prove that \(\lambda_{\max}\leq(n^{1/2}+ p^{1/2})^2+ O(p^{1/2} \log(p))\) (a.e.) for general \(\gamma>0\).

62H10 Multivariate distribution of statistics
62H25 Factor analysis and principal components; correspondence analysis
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