A 2-D realization theory for Markov chains

The dynamics of time-homogeneous Markov chain models is studied from a state-space modeling point of view. It is shown that a Markov chain model can be embedded in a 2-D realization theory where Markov parameters correspond to higher-order transition probabilities. The implication of formulating a Markov chain model in this state-space domain is that many equivalent representations may exist, some of which may have better robustness properties. A modified Hankel approximation algorithm is presented which exactly matches all the Markov parameters. The algorithm is an extension of the 2-D harmonic retrieval algorithm of D.V.B. Rao et al. (1984)