We develop necessary and sufficient conditions for uniqueness of the invariant measure of the filtering process associated to an ergodic hidden Markov model in a finite or countable state space. These results provide a complete solution to a problem posed by Blackwell (1957), and subsume earlier partial results due to Kaijser, Kochman and Reeds. The proofs of our main results are based on the stability theory of nonlinear filters.
"A complete solution to Blackwell’s unique ergodicity problem for hidden Markov chains." Ann. Appl. Probab. 20 (6) 2318 - 2345, December 2010. https://doi.org/10.1214/10-AAP688