  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.
