  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.
