3) the strategy that will lead to the proof of Theorem \[ThJextended\]. We prove the two crucial points needed in Section 4. To make the paper self-contained, we give some background information about different aspects of the problem in the appendices. Several technical issues are also treated there in order to avoid obscuring the proof of the main result.

Outline of Johnstone’s proof
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Before describing the backbone of the proof presented in @imj, we need to introduce a few notational conventions. In what follows, we will use $N$ instead of $p$ to be consistent with the literature. We also denote by ${\mathbf{AB}}$ (for “asymptotic behavior") the situation where $n,N, \text{ and } n/N {\rightarrow}\infty$. We will frequently index functions that depend on both $N$ and $n$ with only $N$. The reason for this is that it will allow us to keep the notations relatively light, and that we think of $n$ as being a function of $N$. Notations like $\textbf{E}_N$ and $\textbf{P}_N$ will denote expectation and probability under the measure induced by the matrices (of size $n(N)\times N$) we are working with.\
Finally, it is technically simpler to work with a matrix $X$ whose entries are standard complex Gaussians 