Mathematics – Probability
Scientific paper
2001-04-10
Mathematics
Probability
This is a preliminary version. Minor misprints are corrected
Scientific paper
Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix $ X^* \* X (X^t \*X) $ converges to the Tracy-Widom law 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 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.$
Soshnikov Alexander
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