integer. For each $1 \leq k \leq m$, let $(\xi_{k,1}, \xi_{k,2}), \zeta_k$ be real random elements that satisfy Assumption \[ass:atom\]. For each $N \geq 1$ and $1 \leq k \leq m$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ real elliptic random matrix with atom variables $(\xi_{k,1}, \xi_{k,2}), \zeta_k$, and assume ${\mathbf}{Y}_{N,1}, \ldots, {\mathbf}{Y}_{N,m}$ are independent. Then the ESD of the product $${\mathbf}{P}_N := N^{-m/2} {\mathbf}{Y}_{N,1} \cdots {\mathbf}{Y}_{N,m}$$ converges almost surely to $F_m$ (with density given by ) as $N \to \infty$.

More generally, we establish a version of Theorem \[thm:simple\] where each elliptic random matrix ${\mathbf}{Y}_{N,k}$ is perturbed by a deterministic, low rank matrix ${\mathbf}{A}_{N,k}$ with small Hilbert-Schmidt norm. In fact, Theorem \[thm:simple\] will follow from Theorem \[thm:main\] below. We recall that, for any $m \times n$ matrix ${\mathbf}{M}$, the Hilbert-Schmidt norm $\|{\mathbf}{M}\|_2$ is given by the formula $$\label{eq:def:hs}
    \|{\mathbf}{M}\|_2 := \sqrt{ \operatorname{tr}({\mathbf}{M} {\mathbf}{M}^\ast) } = \sqrt{ \operatorname{tr}({\mathbf}{M}^\ast {\mathbf}{M})}.$$

\[thm:main\] Let $m > 1$ be an integer. For each $1 \leq k \leq m$, let $(\xi_{k,1}, \xi_{k,2}), \zeta_k$ be real random elements that satisfy Assumption \[ass:atom\]. For each $N \geq 1$ and $1 \leq k \leq m$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ real elliptic random matrix with atom 