Functions of Processes with Markovian States--II

Abstract

In [4] the rank of a state of a stochastic process was defined, although the notion, without the name, is originally due to Gilbert [5]. Let $I = \{1, 2, \cdots\}$. The purpose of the present paper is to prove the THEOREM. Let $\{Y_k\}$ have state space $U_k$ at time $k(k = 1, 2, \cdots$ or $k = 0, \pm 1, \pm 2, \cdots)$. Let $N$ be a finite subset of the index set and, for each $n \varepsilon N$, let $V_n \subset U_n$ be a set of states of finite rank at time $n$. Without loss of generality, assume $(U_n \sim V_n) \mathbf{\cap} (V_n \mathbf{\times} I) = \varnothing$. Then, there exists a process $\{X_k\}$ such that (i) $\{X_k\}$ has state space $U_k$ at time $k \not\in N$ and state space $(U_n \sim V_n) \mathbf{\cup} (V_n \mathbf{\times} I)$ at time $n \varepsilon N$; (ii) The states $(\epsilon, i)$ for $(\mathbf{\epsilon}\varepsilon V_n, i \varepsilon I$ and $n \varepsilon N$ are Markovian; and (iii) $Y_k = F_k (X_k)$ where $F_k(\delta) = \delta$ if $\delta \varepsilon U_k \sim V_k$ (take $V_k = \varnothing$ for $k \not\in N)$ and $F_n (\epsilon, i) = \epsilon$ if $\epsilon\varepsilon V_k$. This theorem is a generalization of Theorem 1 of [4]. Its proof is in Section 2. Section 3 contains corollaries which are the analogues of the corollaries to Theorems 1 and 2 of [4]. These show that if, in addition to the $\epsilon\varepsilon V_n$ for $n \varepsilon N$, there are states of rank 1 (Markovian states) or 2, then $\{X_k\}$ can be constructed so as to preserve the ranks of these states. Section 4 contains a third corollary giving conditions under which $N$ may be infinite. In particular, under these conditions $N$ may be the whole index set and, for each $n \varepsilon N$, we may let $V_n$ be the set of all states of finite rank at time $n$. Corollary 4, also in Section 4, states that, under the conditions of Corollary 3, stationarity in $\{Y_k\}$ may be preserved in $\{X_k\}$. Dharmadhikari [1], [2], [3] has given conditions under which $\{X_k\}$ can be constructed to be a stationary, finite Markov chain. In [2] he requires the condition, among other, that each state of $\{Y_k\}$ be of finite rank. We have weakened our conditions by not insisting on stationarity or finiteness of the state space of $\{Y_k\}$ and by imposing finiteness of rank only on some states. We have completely dropped his condition that certain cones be polyhedral. This last is the reason that we have countably many Markovian states mapping into a single state of $\{Y_k\}$ instead of finitely many.