in Table 1 and the matrices are not as big, but the Tracy-Widom approximation is already acceptable for the upper quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences.

We see that the fit is good to very good for the upper quantiles ($.9$ and beyond) across the range of dimensions we investigated. The practical interest of this remark is clear: these are the quantiles one would naturally use in a testing problem. We note that it appears empirically that the problem gets harder when the ratio $r$ of the larger dimension to the smaller one ($p_1$ in our notation) gets bigger: the larger $r$, the larger $p_1$ should be for the approximation to be acceptable.

Conclusions and Organization
----------------------------

From a technical standpoint, the method developed in @imj proves to be versatile, and, at least conceptually, relatively easy to adapt to the case where $n/p{\rightarrow}\infty$. Nevertheless, substantial technical work is needed to obtain Theorem \[ThJextended\]. Using the elementary fact (see e.g theorem 7.3.7 in @hj) that the largest eigenvalue of $X^{*}X$ is the same as the largest eigenvalue of $XX^{*}$, it will be sufficient to give the proof in the case $n/p{\rightarrow}\infty$.

From a practical point of view, 