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Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the entries of the matrix $X$ are iid, but if you so need to answer the question, you can assume that first, and then perhaps we can see what happens when we put a covariance structure on $X$.

I'd like to know if there're some known results on the distribution of the $k$-th largest eigenvalue of the sample covariance matrix $C:= \frac{1}{p}X^TX$. References to such works would be greatly apprecited, as will be your answer as well!

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    $\begingroup$ You're looking for results on the distribution of the $k$th singular value of $X$. The high dimensional probability book has some results. See chapters 4 and 6.5 in particular where assuming i.i.d. sub-Gaussian they get strong concentration for all singular values, and with weaker distributional assumptions they get sharp expectation bounds (on only the largest singular value). The Marchenko-Pastur distribution is somewhat relevant, but assumes i.i.d. entries. $\endgroup$
    – Mark Schultz-Wu
    Commented Mar 6, 2020 at 7:05
  • $\begingroup$ @Mark Thank you for the book - it's a good on eit seems. I took a quick look at both chapter 4 and chapter 6.5, but I failed to notice any asymtotics which I originally meant to ask. Really my fault that I didn't make this clear - going to ask a separate question. What I'm looking for is the the distribution of the $k$-th largest eigenvalue when both the sample size $n$ and dimension $p$ foes to infinity. $\endgroup$
    – Learning math
    Commented Mar 6, 2020 at 8:47
  • $\begingroup$ @Mark Please see the modified question here where I ask for the asymtotic distribution of the $k$-th singular value - mathoverflow.net/questions/354288/… $\endgroup$
    – Learning math
    Commented Mar 6, 2020 at 9:05

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