On approximation of smoothing probabilities for hidden Markov models

We consider the smoothing probabilities of hidden Markov model (HMM). We show that under fairly general conditions for HMM, the exponential forgetting still holds, and the smoothing probabilities can be well approximated with the ones of double-sided HMM. This makes it possible to use ergodic theorems. As an application we consider the pointwise maximum a posteriori segmentation, and show that the corresponding risks converge.