Asymptotic Synchronization for Finite-State Sources

We extend a recent analysis of synchronization for exact finite-state sources to nonexact sources for which synchronization occurs only asymptotically. Although the asymptotic case requires more sophistication, the primary results remain the same. We find that an observer's average uncertainty in the source state vanishes exponentially fast and, as a consequence, an observer's average uncertainty in predicting future output converges exponentially fast to the source entropy rate.