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

Abstract

Let $\{Y_n, n \geqq 1\}$ 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 $\epsilon$ of $J$, a convex cone $\mathscr{C}(\pi_\epsilon)$. The main theorem (Section 2) asserts that if each $\mathscr{C}(\pi_\epsilon)$ is polyhedral, then $\{Y_n\}$ is a function of a finite Markov chain. The hypothesis that each $\mathscr{C}(\pi_\epsilon)$ 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 $\{Y_n\}$ 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 $\{Y_n\}$ 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 $\{Y_n\}$ need not be irreducible or aperiodic. Our sufficient conditions also do not necessarily yield such a chain.