Sufficient Conditions for a Stationary Process to be a Function of a Finite Markov Chain

Let ₙ, n 1\ be a stationary process with a finite state-space J. We will use the definitions of a function and of a regular function of a finite Markov chain given in [1]. In Section 1 of this paper we define, for each state ε of J, a convex cone C(π_ε). The main theorem (Section 2) asserts that if each C(π_ε) is polyhedral, then ₙ\ is a function of a finite Markov chain. The hypothesis that each C(π_ε) is polyhedral is not quite necessary and some results are given in Section 3 under weaker assumptions. It is also shown that these weaker conditions are necessary for ₙ\ to be a regular function of a finite Markov chain. The final section presents an example which shows that not every function of a finite Markov chain is a regular function of a Markov chain. The results of Gilbert [3] are at the root of our investigation. Gilbert always assumed that the given stationary process ₙ\ and the underlying Markov chain were irreducible and aperiodic. However, his results continue to hold even when these assumptions are dropped. In particular, the results of Section 1 of [3] depend only on the stationary and the Markov character of the underlying chain. Theorem 2 of [3] also holds in a more general set-up (see Lemma 3.1 of [1]). Our stationary process ₙ\ need not be irreducible or aperiodic. Our sufficient conditions also do not necessarily yield such a chain.