If {X-t} is a finite-state Markov process, and {Y-t} is a finite-valued out
put process with Yt+1 depending (possibly probabilistically) on X-t, then t
he process pair is said to constitute a hidden Markov model. This paper con
siders the realization question: given the probabilities of all finite-leng
th output strings, under what circumstances and how can one construct a fin
ite-state Markov process and a state-to-output mapping which generates an o
utput process whose finite-length strings have the given probabilities? Aft
er reviewing known results dealing with this problem involving Hankel matri
ces and polyhedral cones, we develop new theory on the existence and constr
uction of the cones in question, which effectively provides a solution to t
he realization problem. This theory is an extension of recent theoretical d
evelopments on the positive realization problem of linear system theory.