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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, 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!

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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 (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 (2014).

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  • $\begingroup$ Thanks - just accepted your answer! $\endgroup$ – Learning math May 6 '20 at 13:19

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