of a real elliptic random matrix.

In this note, we consider the product ${\mathbf}{Y}_{N,1} \cdots {\mathbf}{Y}_{N,m}$ of $m$ independent real elliptic random matrices. In particular, we assume each real elliptic random matrix ${\mathbf}{Y}_{N,k}$ has atom variables $(\xi_{k,1}, \xi_{k,2}), \zeta_k$ which satisfy the following conditions.

\[ass:atom\] There exists $\tau > 0$ such that the following conditions hold.

1.  $\xi_{k,1}, \xi_{k,2}$ both have mean zero and unit variance.

2.  ${\mathbb{E}}| \xi_{k,1}|^{2 + \tau} + {\mathbb{E}}|\xi_{k,2}|^{2 + \tau} < \infty$. \[item:tau\]

3.  $\rho_k := {\mathbb{E}}[\xi_{k,1} \xi_{k,2}]$ satisfies $|\rho_k| < 1$. \[item:rhoass\]

4.  $\zeta_k$ has mean zero and finite variance.

In our main result below, we show that the limiting distribution $F_m$ (with density given by ) from Theorem \[thm:GTprod\] for the product of independent iid random matrices is also the limiting distribution for the product of independent elliptic random matrices. In other words, the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. This type of universality was first considered by Burda, Janik, and Waclaw in [@BJW] for matrices with Gaussian entries; see also [@Bsurv]. Figure \[fig:plots\] displays several numerical simulations which illustrate this phenomenon.

\[thm:simple\] Let $m > 1$ be an 