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

• Why not using free probability? Solving problems with prescribed methods rather belongs to philosophy, and that's far away from RMT, which is applied math. – Richard Apr 9 '19 at 15:20
• I know how to solve it using FPT, I was interested to know if there were a simpler method. – user40130 Feb 23 '20 at 23:29

Wouldn't it be helpful, if $$\alpha$$ and $$\beta$$ are positive to use the fact that $$\mathbf{M}$$ has the same law than $$\mathbf{H}\Sigma\mathbf{H}^{*}$$ with $$\mathbf{H}$$ the $$N\times 2N$$ matrix filled with independant entries of law $${\mathcal{N}}(\mu=0 ,\sigma^{2}=\frac{1}{N})$$ and $$\Sigma$$ the $$2N$$ diagonal matrix $$Diag(\alpha,\dots,\alpha,\beta,\dots,\beta)$$ ?
Then I think that most classical method to get the MP law can be used on $$\mathbf{M}$$.
• Thank you for this. Yes my thoughts were that it could be solved using this decomposition, however I am unclear on the details (I am not familiar with generalizing the MP law). I can see how to find the AED for $\mathbf{H}\mathbf{H}^{*}$ in your solution, but I don't know how that extends to $\Sigma\mathbf{H}\mathbf{H}^{*}$. Presumably you just weight using $\alpha$ and $\beta$ and normalize using $N$? Do you have a reference or exercise that uses the classical method you mention? – user40130 Feb 23 '20 at 23:40