$F_2$ (with density given by when $m=2$) as $N \to \infty$.

Overview and outline
--------------------

We begin by outlining the proof of Theorem \[thm:main\]. Instead of directly considering $ {\mathbf}{P}_N := N^{-m/2} \prod_{k=1}^m ( {\mathbf}{Y}_{N,k} + {\mathbf}{A}_{N,k})$, we introduce a linearized random matrix, ${\mathbf}{Z}_N:= \frac{1}{\sqrt{N}} ({\mathbf}{Y}_N + {\mathbf}{A}_N)$, where ${\mathbf}{Y}_N$ and ${\mathbf}{A}_N$ are $mN \times mN$ block matrices of the form $$\label{def:YN}
    {\mathbf}{Y}_N := \begin{bmatrix} 
                0 &  {\mathbf}{Y}_{N,1} &     &            & 0       \\
                                0 & 0    & {\mathbf}{Y}_{N,2} &            & 0        \\      
                                &      & \ddots & \ddots     &         \\
                                0 &      &     &          0 & {\mathbf}{Y}_{N,m-1} \\
                            {\mathbf}{Y}_{N,m} &      &     &            &  0      
    \end{bmatrix}$$ and $$\label{def:AN}
    {\mathbf}{A}_N := \begin{bmatrix} 
                0 &  {\mathbf}{A}_{N,1} &     &            & 0       \\
                                0 & 0    & {\mathbf}{A}_{N,2} &            & 0        \\      
                                &      & \ddots & \ddots     &         \\
                                0 &      &     &          0 & {\mathbf}{A}_{N,m-1} \\
                            {\mathbf}{A}_{N,m} &      &     &            &  0      
            \end{bmatrix}.$$

The following theorem gives the limiting distribution of ${\mathbf}Z_N$, from which we will deduce our main theorem as a corollary.

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

In Section \[sec:linear\], we 