Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here a modified question.
I'm new to random matrix theory, but per my understanding Tracy-Widom law describes the the asymptotic distribution of the largest eigenvalue of any square real symmetrix matrix with iid entries on the diagonal and above it, of dimension $n \times n$ as $n \to \infty$. (Please correct me if I'm wrong!).
What I'm trying to do is to connect, or at least find a resource that connect Tracy-Widom with Marcenko-Pastur law in a somewhat detailed way, as follows.
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 $X$. Let, as often in the random matrix domain, let $n, p \to \infty, p/n \to c \in (0, \infty)$.I'm interested in the limiting distribution of $k$-th largest eigenvalue, when $k$ is fixed and $k$ - is varying.
Precisely, my questions are:
(1) What's the limiting distruition of the $k$-th largest eigenvalue of $\frac{1}{p}XX^{T}$, as $p, n \to \infty, p/n \to c?$
(2) Also what's the limiting distribution of the $k$-th largest eigenvalue of $\frac{1}{p}XX^{T}$, as $k, p, n \to \infty, p/n \to c, k/p \to c', c\in (0, \infty), c' \in (0,1) ?$
To show you an idea what I'm after, I'll mentiong what I found from my search:
(1)I found this paper that seems to be relevant: https://projecteuclid.org/download/pdfview_1/euclid.aoap/1481792600, but they deal with the the limiting distribution of the largest eigenvalue.
(2) This paper by Tracy and Widom: https://arxiv.org/pdf/hep-th/9211141.pdf, describes in Section E the probability density for the $k$-th largest eigenvalue. But I think there the underlying matrix is a real (or complex) symmetric (or Hermitian) matrix with iid entries on the diagonal and above the diagonal, and not sample covariance matrix.
Any help will be sincerely appreciated, as I'm super new to RMT!
