a complex random variable with mean zero and unit variance. For each $N \geq 1$, let ${\mathbf}{Y}_N$ be a $N \times N$ iid random matrix with atom variable $\xi$. Then the ESD of $\frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges almost surely to the circular law $F_{\mathrm{circ}}$ as $N \rightarrow \infty$.

More recently, Götze and Tikhomirov [@GTprod] consider the ESD of the product of $m$ independent iid random matrices. They show that, as the sizes of the matrices tend to infinity, the limiting distribution is given by $F_m$, where $F_m$ is supported on the unit circle in the complex plane and has density $f_m$ given by $$\label{plotnost}
    f_m(z) := \left\{
                \begin{array}{ll}
                    \frac{1}{m \pi} |z|^{\frac{2}{m}-2}, & \text{ for } |z|\leq 1 \\
                    0, & \text{ for } |z| > 1
                \end{array} \right.$$ in the complex plane. It can be verified directly, that if $\psi$ is a random variable distributed uniformly on the unit disk in the complex plane, then $\psi^m$ has distribution $F_m$.

\[thm:GTprod\] Let $m \geq 1$ be an interger, and assume $\xi_1, \ldots, \xi_m$ are complex random variables with mean zero and unit variance. For each $N \geq 1$ and $1 \leq k \leq m$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ iid 