  The EM procedure is a principal tool for parameter estimation in the hidden
Markov models. However, applications replace EM by Viterbi extraction, or
training (VT). VT is computationally less intensive, more stable and has more
of an intuitive appeal, but VT estimation is biased and does not satisfy the
following fixed point property. Hypothetically, given an infinitely large
sample and initialized to the true parameters, VT will generally move away from
the initial values. We propose adjusted Viterbi training (VA), a new method to
restore the fixed point property and thus alleviate the overall imprecision of
the VT estimators, while preserving the computational advantages of the
baseline VT algorithm. Simulations elsewhere have shown that VA appreciably
improves the precision of estimation in both the special case of mixture models
and more general HMMs. However, being entirely analytic, the VA correction
relies on infinite Viterbi alignments and associated limiting probability
distributions. While explicit in the mixture case, the existence of these
limiting measures is not obvious for more general HMMs. This paper proves that
under certain mild conditions, the required limiting distributions for general
HMMs do exist.
