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, or also that $x\_i$ are of the form $x\_i = C^{1/2}z\_i$, where $Z:=[z\_1 \dots z\_n]$ have iid entries (I think the work of Baik and Silverstein includes this case for ESD's). I'm really assuling that the co-ordinates ("features") of each $x\_i$ are corelated.


I'm wondering (some references will be enough, but detailed answers appreciated!) if there're equivalents of Marcenko-Pastur and Tracy-Wisdom theorems in this case? More precisely:


(1) What's the limiting empirical spectral distribution of sample covariance $\frac{1}{p}XX'$ and Gram matrix $\frac{1}{n}X'X$?


(2) How are the largest eigenvalues of the above two matrices distributed? 


Thank you for your answer!



Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references
If the correlations can be described by a multivariate Gaussian there exist results for the spectral density in the limit of large matrices, see for example [Spectral Moments of Correlated Wishart Matrices](https://www.researchgate.net/publication/7952513_Spectral_Moments_of_Correlated_Wishart_Matrices) (2005).


For the distribution of the largest eigenvalue I only know results for complex matrix elements, see [Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges](https://arxiv.org/abs/1409.7548) (2014).


