Identification of State-calculable Functions of Finite Markov Chains

Abstract

If $\{f(X_n): n = 1, 2, \cdots \}$ is a finite-state function of a finite-state Markov chain $\{X_n\}$, it is known that the distribution of $\{f(X_n)\}$ is determined by the distribution of $(f(X_1), f(X_2), \cdots, f(X_K))$ for suitable $K$, and a finite construction utilizing only the latter distribution exists in special cases yielding the probability structure of a chain $\{X'_n\}$ and a function $f'$ such that $\{f(X_n\}$ and $\{f'(X'_n)\}$ have the same distribution [7]. We obtain a finite construction when $(\{X_n\}, f)$ is such that if $i$ is a state of $\{X_n\}$ and $j$ is a state of $\{f(X_n)\}$, then there is at most one transition from $i$ to the set $f^{-1}(j)$ and the distribution of $X_1$ assigns positive probability to at most one state in each set $f^{-1}(j)$. Such "state-calculability" is a rather severe, but natural, structural restriction subsuming certain cases not previously treated. The corresponding finite construction is very simple and directly related to the representation of any finite-state process $\{Y_n\}$ by a function of a (possibly countable-state) Markov chain.