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 β in some ratio n/pβΞ³>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 nβp=O(p1/3). We also prove that
\lambda_{max} \leq (n^{1/2}+p^{1/2})^2 +O(p^{1/2}\*\log(p)) (a.e.) for
general Ξ³>0.Comment: This is a preliminary version. Minor misprints are correcte