  The infinite Viterbi alignment is the limiting maximum a-posteriori estimate
of the unobserved path in a hidden Markov model as the length of the time
horizon grows. For models on state-space $\mathbb{R}^{d}$ satisfying a new
``decay-convexity'' condition, we develop an approach to existence of the
infinite Viterbi alignment in an infinite dimensional Hilbert space.
Quantitative bounds on the distance to the infinite Viterbi alignment, which
are the first of their kind, are derived and used to illustrate how approximate
estimation via parallelization can be accurate and scaleable to
high-dimensional problems because the rate of convergence to the infinite
Viterbi alignment does not necessarily depend on $d$. The results are applied
to approximate estimation via parallelization and a model of neural population
activity.
