## The Annals of Probability

### Countable State Space Markov Random Fields and Markov Chains on Trees

Stan Zachary

#### Abstract

Let $S$ and $A$ be countable sets and let $\mathscr{G}(\Pi)$ be the set of Markov random fields on $S^A$ (with the $\sigma$-field generated by the finite cylinder sets) corresponding to a specification $\Pi$, Markov with respect to a tree-like neighbour relation in $A$. We define the class $\mathscr{M}(\Pi)$ of Markov chains in $\mathscr{G}(\Pi)$, and generalise results of Spitzer to show that every extreme point of $\mathscr{G}(\Pi)$ belongs to $\mathscr{M}(\Pi)$. We establish a one-to-one correspondence between $\mathscr{M}(\Pi)$ and a set of "entrance laws" associated with $\Pi$. These results are applied to homogeneous Markov specifications on regular infinite trees. In particular for the case $|S| = 2$ we obtain a quick derivation of Spitzer's necessary and sufficient condition for $|\mathscr{G}(\Pi)| = 1$, and further show that if $|\mathscr{M}(\Pi)| > 1$ then $|\mathscr{M}(\Pi)| = \infty$.

#### Article information

Source
Ann. Probab., Volume 11, Number 4 (1983), 894-903.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993439

Digital Object Identifier
doi:10.1214/aop/1176993439

Mathematical Reviews number (MathSciNet)
MR714953

Zentralblatt MATH identifier
0524.60056

JSTOR