Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where:

$\mathbf{X}_i=\mathbf{H}_i\mathbf{H}_i^{*}$ for $i\in\{1,2\}$ where $\mathbf{H}_i$ are both square $N\times N$ random matrices whose entries are i.i.d and follow a normalised Gaussian distribution $\sim {\mathcal{N}}(\mu=0 ,\sigma^{2}=\frac{1}{N})$, so that the asymptotic eigenvalue distributions of $\mathbf{X}_i$ are given by the Marchenko-Pastur law with $\beta=1$.

More generally, is there such a method for finding the AED of the linear combination $\mathbf{M}_p=\alpha\mathbf{X}_1+ \beta\mathbf{X}_2$ for $\alpha, \beta \in \mathbb{R}$?

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    $\begingroup$ I would suggest you ask this question in Mathoverflow. I think this question is quite advanced to get good answers on this site. $\endgroup$ – Landon Carter Oct 1 '18 at 17:03
  • $\begingroup$ Thanks, I'll do that. $\endgroup$ – user40130 Oct 2 '18 at 14:09

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