Cesàro summability in a linear autonomous difference equation

For a linear autonomous difference equation with a unique real eigenvalue  $\lambda _{0}$, it is shown that for every solution $x$ the ratio of $x$ and the eigensolution corresponding to  $\lambda _{0}$ is Cesàro summable to a limit which can be expressed in terms of the initial data. As a consequence, for most solutions the Lyapunov characteristic exponent is equal to  $\lambda _{0}$. The proof is based on a Tauberian theorem for the Laplace transform.