Suppose {1 , ... , m} is a finite set, and {Y_t} is a stationary stochastic process assuming values in this finite set. In this talk we address the so-called "partial realization problem" for this process. Suppose k is a given finite integer, and we wish to construct a hidden Markov model (HMM) that perfectly reproduces the frequencies of all k-tuples, or a k-th order partial realization. We present a systematic way of constructing such partial realizations. In particular, we show that a well-known realization of the process in terms of a (k-1)-st order Markov process is in fact the only solution to the partial realization problem if we impose one additional condition. These results justify the widespread use of multi-step Markov models in partial realization.
The talk will be completely self-contained in terms of defining all the terms used and will review all relevant previously known results.
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