large - is nowadays a widely used tools in many fields, such as image analysis, signal processing, functional data analysis and quantitative finance. Several examples come to mind, including Eigenfaces, subspace filtering, or @lalouxetal where PCA (as well as some random matrix theory) is used to try to improve on the naive solution to Markovitz’s portfolio optimization problem.

Important progress has been made recently in our understanding of the statistical properties of PCA in such settings. Emblematic of this is work of @imj, which explains the properties of the square of the largest singular value of a random matrix $X$ under the “null model" where its entries are iid ${\cal N}(0,1)$. Specifically, if we denote the sample eigenvalues of $X'X$ by $l_1\geq \ldots \geq l_p$, call $$\begin{aligned}
n_1&=\max{(n,p)}-1\;, \; \; \; \; p_1=\min{(n,p)} \;,\\
\mu_{np}&=(\sqrt{n_1}+\sqrt{p_1})^2 \; ,\\
\sigma_{np}&=(\sqrt{n_1}+\sqrt{p_1})\left(\frac{1}{\sqrt{n_1}}+\frac{1}{\sqrt{p_1}}\right)^{1/3} \; ,\end{aligned}$$ and $W_1$ the Tracy-Widom distribution (see **A0**), it was shown in @imj that

\[ThJ\] If $n,p {\rightarrow}\infty$ and $n/p{\rightarrow}\gamma \in (0,\infty)$, $$\frac{l_1-\mu_{np}}{\sigma_{np}} \overset{\cal L}{\rightarrow}W_1 \;.$$

Building on @imj and using properties of determinantal point processes, @sosh showed that the same result holds for the $k$ largest eigenvalues, where $k$ is a fixed integer: their joint distribution converges to their Tracy-Widom 