Some Regular and Non-regular Functions of Finite Markov Chains

Abstract

Let $J$ be a finite non-empty set and let $S(J)$ denote the set of all finite sequences of elements of $J$. If $s = (\delta_1, \cdots, \delta_m)\in S(J)$ and $t = (\mu_1, \cdots, \mu_n)\in S(J)$, then $st$ will denote the combined sequence $(\delta_1, \cdots, \delta_m, \mu_1, \cdots, \mu_n)$. The singleton sequence $(\delta)$ will be denoted by $\delta$. The symbol $s^2$ will mean the sequence $ss$ and the symbols $s^3,s^4$, etc., are defined similarly. Suppose $\{Y_n\}$ is a stationary process with state-space $J$. If $s \in S(J)$ and has length $n, p(s)$ denotes $P\lbrack (Y_1, \cdots, Y_n) = s \rbrack$. The rank $n(\delta)$ of a $\delta \in J$ is defined to be the largest integer $n$ such that we can find $2n$ sequences $s_1, \cdots, s_n, t_1, \cdots, t_n$ in $S(J)$ such that the $n \times n$ matrix $\|p(s_i\delta t_j)\|$ is non-singular. Suppose now that $\{Y_n\}$ is a function of a finite Markov chain (hereafter abbreviated ffMc). That is, let there exist a stationary Markov chain $\{X_n\}$ with a finite state-space $I$ and a function $f$ on $I$ onto $J$ such that $\{Y_n\}$ and $\{f(X_n)\}$ have the same distribution. Then Gilbert [5] has shown that $n(\delta) \leqq N(\delta)$ for all $\delta \in J$, where $N(\delta)$ is the number of elements in $f^{-1}\lbrack\{\delta\}\rbrack$. If we can find $\{X_n\}$ and $f$ in such a way that $n(\delta) = N(\delta)$ for all $\delta \in J$, then $\{Y_n\}$ is said to be a regular ffMc. The motivation for investigating the regularity property of a ffMc has been made clear by Gilbert in the first and the last paragraphs of Section 2 of [5]. Fox and Rubin [3] have given an example of a process $\{Y_n\}$ which has $n(\delta) < \infty$ for all $\delta \in J$ but which is not a ffMc. In the first part of this paper we expand their example into a class of examples and show that some of these examples yield nonregular ffMc. These examples are of a different nature than those given in [1], Section 4. Further our method of investigation is different from that employed by Fox and Rubin. The second part of this paper is devoted to proving that an exchangeable process which is a ffMc is a regular ffMc.