Two-Sided Markov Chains

Abstract

Let $S$ be a countable set, and $a, b$ two distinct elements not in $S$. Let $\Omega$ be the set of functions $\omega: Z$ (integers) $\rightarrow S \cup \{a, b\}$ so that (i) $\omega (n) \in S$ for some $n$, (ii) $\omega (n) = a \Rightarrow \omega(m) = a$ for $m < n$, and (iii) $\omega(n) = b \Rightarrow \omega(m) = b$ for $m > n$. On $\Omega$, let $x(n)$ be the $n$th coordinate, $\theta_n$ the shift operator, and $\alpha = \inf\{n: x(n) \in S\}$. A measure $P$ on $\Omega$ is Markov if $\forall i \in S, n \in Z, P(x(n) = i) < \infty$, and $P(x(n + 1) = i\mid \sigma(x(k): k \leq n)) = P(x(n + 1) = i|x(n))$ on $\{x(n) \in S\}$. A Markov measure $P$ is called a two-sided Markov chain if there are substochastic matrices on $S, p$ and $q$, so that $\forall i, j \in S, n \in Z, P(x(n) = i, x(n + 1) = j) = P(x(n) = i)p(i, j) = P(x(n + 1) = j)q(j, i)$. It is shown that if $p$ is a certain kind of irreducible matrix then there exists an integer $d \geq 1$ and a real $\rho > 0$ so that $P \circ \theta^{-1}_{nd} = \rho^nP \forall n \in Z$. Of particular interest is the generalization of the case $d = 1$. A two-sided Markov chain is called quasistationary if there exists $\rho > 0$ with $P \circ \theta^{-1}_n = \rho^n P$. It is shown that if $\pi$ is a measure on $S$ and $\rho > 0$, and $p$ is a substochastic matrix on $S$, with $\pi p \leq \rho \pi$, then there exists a quasistationary chain with $p$ as forward transition, $P(x(0) = i) = \pi(i),$ and $P \circ \theta^{-1}_n = \rho^n P. P$ is used to prove the Riesz decomposition theorem for $\pi$. Finally, it is shown under certain verifiable assumptions, that a quasistationary chain, restricted to $\{\alpha \leq 0\}$, is an extended chain, with transition $p$, which is a certain kind of approximate $p$-chain in the sense of Hunt.