On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity

TL;DR

This paper extends the Tracy-Widom law applicability for the largest eigenvalue of Wishart matrices with identity covariance to cases where the ratio of dimensions tends to zero or infinity, supported by theoretical proofs and numerical experiments.

Contribution

It proves that the Tracy-Widom law holds for the largest eigenvalue even when p/n or n/p tends to infinity, broadening previous results.

Findings

Tracy-Widom law applies in more extreme dimension ratios.

The joint distribution of top eigenvalues follows Tracy-Widom.

Numerical results confirm the approximation's accuracy in small dimensions.

Abstract

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".