# Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?

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 $$\alpha, \beta \in \mathbb{R}$$?

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