Existence and Uniqueness of Countable One-Dimensional Markov Random Fields

Abstract

For a countable set $S$ and strictly positive matrix $Q = (Q(x, y))_{x, y\in S}$ let $\mathscr{G}(Q)$ be the set of all probability measures $\mu$ on $\Omega \equiv S^\mathbb{Z}$, strictly positive on cylinder sets, and with the following "two-sided Markov property": $\mu \{\omega_n = x\mid\omega_l, l \neq n\} = \lbrack Q^2(x, z) \rbrack^{-1}Q(y, x)Q(x, z)$ a.e. on the set $\{\omega_{n - 1} = y, \omega_{n + 1} = z\}$. In other words, for every $\mu\in\mathscr{G}(Q)$, the conditional distribution of $\omega_n$ given all other $\omega_l$ depends on $\omega_{n - 1}$ and $\omega_{n + 1}$ only, and "behaves as if $\{\omega_n\}_{n\in \mathbb{Z}}$ is a Markov chain with transition probability matrix $Q$." $\mathscr{G}_0(Q)$ denotes the set of those $\mu\in \mathscr{G}(Q)$ which are in addition translation invariant. We establish a conjecture of Spitzer's [9] that either $\mathscr{G}_0(Q) = \varnothing$ or $\mathscr{G}_0(Q)$ consists of one element only, which is then necessarily a stationary Markov chain on $\Omega$. We also give a condition for $\mathscr{G}(Q) = \varnothing$.