  Consider a filtering process associated to a hidden Markov model with
densities for which both the state space and the observation space are
complete, separable, metric spaces. If the underlying, hidden Markov chain is
strongly ergodic and the filtering process fulfills a certain coupling
condition we prove that, in the limit, the distribution of the filtering
process is independent of the initial distribution of the hidden Markov chain.
If furthermore the hidden Markov chain is uniformly ergodic, then we prove that
the filtering process converges in distribution.
