the constants $C$ and $K$ to depend on the integer $m$ from Theorem \[thm:main\]; we will not denote this dependence with a subscript.

In view of Theorem \[thm:main\] and Assumption \[ass:atom\], we define the correlations $\rho_k := {\mathbb{E}}[ \xi_{k,1} \xi_{k,2} ]$ for the atom variables $\xi_{k,1}, \xi_{k,2}$. In addition, we let $\tau > 0$ be such that $$\sum_{k=1}^m \left( {\mathbb{E}}|\xi_{k,1}|^{2+\tau} + {\mathbb{E}}|\xi_{k,2}|^{2+\tau} \right) < \infty.$$

Proof of Theorem \[thm:main\]  {#sec:linear}
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A matrix-valued Stieltjes transform {#sec:resolvent}
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Least singular value bound {#sec:singularvalue}
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Completing the argument {#sec:complete}
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Proof of Theorem \[thm:prodWigner\] {#sec:Wigner}
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This section is devoted to the proof of Theorem \[thm:prodWigner\]. As noted above, the proof of Theorem \[thm:prodWigner\] is very similar to the proof of Theorem \[thm:main\]. We define the linearized matrix ${\mathbf}{Z}_N := \frac{1}{\sqrt{N}} {\mathbf}{Y}_N$, where $${\mathbf}{Y}_N := \begin{bmatrix} 0 & {\mathbf}{Y}_{N,1} \\ {\mathbf}{Y}_{N,2} & 0 \end{bmatrix}.$$

As before, the proof of Theorem \[thm:prodWigner\] reduces to showing that the ESD of ${\mathbf}{Z}_N$ converges to the circular law as $N$ tends to infinity.

\[thm:circularWigner\] Under the assumptions of Theorem \[thm:prodWigner\], the ESD of ${\mathbf}{Z}_N := \frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges almost surely to the circular law $F_{\mathrm{circ}}$ as $N \to \infty$.

The proof that Theorem \[thm:prodWigner\] follows from 