Suppose that $$L=\prod \limits_{i=1}^n M_i,$$ where $M_i$ are $2\times2$ matrices of the form $$M_i=\begin{bmatrix} \alpha_1\exp(j\phi_{1,i}) & \alpha_2\exp(j\phi_{2,i}) \\ \alpha_2\exp(j\phi_{1,i}) & \alpha_1\exp(j\phi_{2,i}) \end{bmatrix}=\begin{bmatrix} \alpha_1 & \alpha_2 \\ \alpha_2 & \alpha_1 \end{bmatrix}\times\begin{bmatrix} \exp(j\phi_{1,i}) & 0 \\ 0 &\exp(j\phi_{2,i}) \end{bmatrix},$$ where $\phi_{1,i}$ and $\phi_{2,i}$ are chosen i.i.d. with Uniform distribution over $[0,2\pi]$ and \begin{align*} D=\begin{bmatrix} \alpha_1 & \alpha_2 \\ \alpha_2 & \alpha_1 \end{bmatrix} \end{align*} is a Unitary Matrix.
What can we say about the distribution of $L$, when $n \rightarrow \infty$? Can a variant of Central Limit Theorem be applied here? Or at least an upper bound on the maximum probability of entries of $L$?