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).