  Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix
X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of
particular interest for principal component analysis. For this model, when n, p
tend to infinity and n/p tends to gamma in (0,\infty), it was shown in
Johnstone (2001) that l_1, properly centered and scaled, converges to the
Tracy-Widom law. We show that with the same centering and scaling, the result
is true even when p/n or n/p tends to infinity. The derivation uses ideas and
techniques quite similar to the ones presented in Johnstone (2001). Following
Soshnikov (2002), we also show that the same is true for the joint distribution
of the k largest eigenvalues, where k is a fixed integer. Numerical experiments
illustrate the fact that the Tracy-Widom approximation is reasonable even when
one of the dimension is "small".
