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 applications we consider the pointwise maximum a posteriori segmentation, and show that the corresponding risks converge.

Since the early days of digital communication, Hidden Markov Models (HMMs) have now been routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. An HMM $(X_i,Y_i)_{i\ge 1}$ assumes observations $X_1,X_2,...$ to be conditionally independent given an "explanotary" Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given the observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also called Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments of HMMs with two hidden states when $n$ tends to infinity. It has indeed been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions. This work proves the existence of infinite Viterbi alignments for virtually any HMM with two hidden states.

Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM $(X_i,Y_i)_{i\ge 1}$, observations $X_1,X_2,...$ are assumed to be conditionally independent given an ``explanatory'' Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} (MAP) estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also known as Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments when $n\to \infty$. Thus, it has been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more constructive manner and for a very general class of HMMs.