counterpart.

This is a very interesting development because the classical theory (e.g @anderson) was developed under the assumption that $p$ was fixed and $n$ grew to $\infty$, whereas in modern day applications both $p$ and $n$ are large. However, Johnstone’s assumption $n/p{\rightarrow}\gamma$ imposes a limit on the validity of his result which one would like to remove. In an actual data analysis, with given $p$ and $n$, $n={\mathrm{o}}(p)$ and $n\asymp p$ could be equally plausible. Furthermore, a specific $X$ of size $n\times p$ could arise in many triangular arrays settings, where we have $X_j$ of size $n_j\times p_j$, and the limitation $n_j/p_j{\rightarrow}\gamma$ finite might only hold in some triangular situations and not in others.

Accordingly in this paper we weaken the assumption that $n/p {\rightarrow}\gamma$ finite and show that

\[ThJextended\] If $n,p {\rightarrow}\infty$ and $n/p{\rightarrow}\infty$, $$\frac{l_1-\mu_{np}}{\sigma_{np}} \overset{\cal L}{\rightarrow}W_1 \;.$$

Moreover, with the same centering and scaling, the joint distribution of the $k$ largest eigenvalues converges in law to its Tracy-Widom counterpart.

Dually, the same result holds if $n/p{\rightarrow}0$.

Let us note that the remark we made about centering and scaling sequences after Theorem \[ThJ\] is still valid in this context.

There is clearly a mathematical motivation for dealing with this 