random matrix with atom variable $\xi_k$, and assume ${\mathbf}{Y}_{N,1}, \ldots, {\mathbf}{Y}_{N,m}$ are independent. Define the product $${\mathbf}{P}_N := N^{-m/2} {\mathbf}{Y}_{N,1} \cdots {\mathbf}{Y}_{N,m}.$$ Then ${\mathbb{E}}F^{{\mathbf}{P}_N}$ converges to $F_m$ as $N \to \infty$.

The convergence of $F^{{\mathbf}{P}_N}$ to $F_m$ in Theorem \[thm:GTprod\] was strengthened to almost sure convergence in [@B; @OS]. The Gaussian case was originally considered by Burda, Janik, and Waclaw [@BJW]; see also [@Bsurv]. We refer the reader to [@AB; @ABK; @AIK; @AIK2; @AKW; @AS; @BJLNS; @F; @F2] and references therein for many other interesting results concerning products of Gaussian random matrices.

New results
===========

In this paper, we generalize Theorem \[thm:GTprod\] by considering products of independent real elliptic random matrices. Elliptic random matrices were originally introduced by Girko [@Gorig; @Gten] in the 1980s.

\[def:elliptic\] Let $(\xi_1, \xi_2)$ be a random vector in $\mathbb{R}^2$, and let $\zeta$ be a real random variable. We say ${\mathbf}{Y}_N = (y_{ij})_{i,j=1}^N$ is a $N \times N$ *real elliptic random matrix* with atom variables $(\xi_1,\xi_2), \zeta$ if the following conditions hold.

-   (independence) $\{ y_{ii} : 1 \leq i \leq N\} \cup \{ (y_{ij}, y_{ji}) : 1 \leq i < j \leq N\}$ is a collection of independent random elements.

-   (off-diagonal entries) $\{ (y_{ij}, 