things, the perturbation by ${\mathbf}{A}_{N,k}$ in Theorem \[thm:main\] allows one to consider elliptic random matrices with nonzero mean. Indeed, let $\mu_k$ be a real number, and assume each entry of ${\mathbf}{A}_{N,k}$ takes the value $\mu_k$. Then ${\mathbf}{Y}_{N,k} + {\mathbf}{A}_{N,k}$ is an elliptic random matrix whose atom variables have mean $\mu_k$.

In [@GNT], a result similar to Theorem \[thm:simple\] is proved under a different set of assumptions.

As noted above, when $\rho_k = 1$, the matrix ${\mathbf}{Y}_{N,k}$ is known as a real symmetric Wigner matrix. Theorem \[thm:main\] requires that $|\rho_k| < 1$, but in the special case when $m=2$, we are able to extend our proof to show that the same result holds for the product of two independent real symmetric Wigner matrices.

\[thm:prodWigner\] Let $\xi_{1,1}, \xi_{2,1}$ be real random variables with mean zero and unit variance, and which satisfy $${\mathbb{E}}|\xi_{1,1}|^{2+\tau} + {\mathbb{E}}|\xi_{2,1}|^{2+\tau} < \infty$$ for some $\tau > 0$. For each $N \geq 1$ and $k=1,2$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ real symmetric matrix whose diagonal entries and upper diagonal entries are iid copies of $\xi_{k,1}$, and assume ${\mathbf}{Y}_{N,1}$ and ${\mathbf}{Y}_{N,2}$ are independent. Then the ESD of the product $${\mathbf}{P}_N := N^{-1} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2}$$ converges almost surely to 