  Random non-Hermitian Jacobi matrices $J_n$ of increasing dimension $n$ are
considered. We prove that the normalized eigenvalue counting measure of $J_n$
converges weakly to a limiting measure $\mu$ as $n\to\infty$. We also extend to
the non-Hermitian case the Thouless formula relating $\mu$ and the Lyapunov
exponent of the second-order difference equation associated with the sequence
$J_n$. The measure $\mu$ is shown to be log-H\"older continuous.
