  For suitable classes of random Verblunsky coefficients, including
independent, identically distributed, rotationally invariant ones, we prove
that if \[ \mathbb{E} \biggl(\int\frac{d\theta}{2\pi}
\biggl|\biggl(\frac{\mathcal{C} + e^{i\theta}}{\mathcal{C} -e^{i\theta}}
\biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 |k-\ell|} \] for some
$\kappa_1 >0$ and $p<1$, then for suitable $C_2$ and $\kappa_2 >0$, \[
\mathbb{E} \bigl(\sup_n |(\mathcal{C}^n)_{k\ell}|\bigr) \leq C_2 e^{-\kappa_2
|k-\ell|} \] Here $\mathcal{C}$ is the CMV matrix.
