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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 $$\gamma>0$$.

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