## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 4 (1976), 557-569.

### 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 $\mathscr{G}(Q) = \varnothing$.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 4 (1976), 557-569.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996027

**Digital Object Identifier**

doi:10.1214/aop/1176996027

**Mathematical Reviews number (MathSciNet)**

MR410930

**Zentralblatt MATH identifier**

0367.60080

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A25

**Keywords**

Markov random field translation invariant Markov random field stationary Markov chain strong ratio limit property

#### Citation

Kesten, Harry. Existence and Uniqueness of Countable One-Dimensional Markov Random Fields. Ann. Probab. 4 (1976), no. 4, 557--569. doi:10.1214/aop/1176996027. https://projecteuclid.org/euclid.aop/1176996027