The Realization Problem for Hidden Markov Models: The Complete Realization Problem

Suppose m is a positive integer, and let M:= {1,...,m}. Suppose {Yt} is a stationary stochastic process assuming values in M. IN this paper we study the question: When does there exist a hidden Markov model (HMM) that perfectly reproduces the complete statistics of this process? Though HMM's are more than forty years old, no complete solution to this problem is available. It is known that a necessary condition for the process to have a HMM is that an assoicated `Hankel' matrix should have finite rank. It is also known that the condition is not sufficient in general. In subsequent work, an alogrithm for constructing a HMM for a finite rank process has been given, assuming at the outset that the process has a HMM. Hence, to date there are no conditions, either necessary or sufficient, for a process to have a HMM that can be stated in terms of the process alone, and nothing else. Against this background, in the present paper we show the following: (i) Suppose a process has finite Hankel rank. Then there always exists a `regular quasi-realization' of the process. Moreover, two regular quasi-realizations are related through a similarity transformation. (ii) If in addition the process is á- mixing, every regular quasi-realization has additional features. Specifically, the `state transition' matrix associated with the quasi-realization satisfies the `quasi-strong Perron property' (its spectral radius is one, the spectral radius is a simple eigenvalue, and there are no other eigenvalues on the unit circle).