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 (Xi,Yi)i≥1 assumes
observations X1,X2,... to be conditionally independent given an
"explanotary" Markov process Y1,Y2,..., which itself is not observed;
moreover, the conditional distribution of Xi depends solely on Yi.
Central to the theory and applications of HMM is the Viterbi algorithm to find
{\em a maximum a posteriori} estimate q1:n=(q1,q2,...,qn) of Y1:n
given the observed data x1: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.Comment: Several minor changes and corrections have been made in the arguments
as suggested by anonymous reviewers, which should hopefully improve
readability. Abstract has been adde