The Number of Phases of Inhomogeneous Markov Fields with Finite State Spaces on N and Z and their Behaviour at Infinity

Abstract

AbstractWe consider the set 𝒢 of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces En, n ϵ T, with fixed local characteristics. For T = N we show that 𝒢 has at most \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_\infty = \mathop {\lim \inf}\limits_{n \to \infty} \left| {\mathop E\nolimits_n} \right| $\end{document} phases. If T = Z, 𝒢 has at most N‐∞ · N∞; phases, where \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_{-\infty} = \mathop {\lim \inf}\limits_{n \to -\infty} \left| {\mathop E\nolimits_n} \right| $\end{document}. We give examples, that for T = N for any number k, 1 ≦ k ≦ N∞, there are local characteristics with k phases, whereas for T = Z every number l · k, 1 ≦ l ≦ N‐∞, 1 ≦ k ≦ N∞ occurs. We describe the inner structure of 𝒢, the behaviour at infinity and the connection between the one‐sided and the two‐sided tail‐fields. Simple examples of Markov fields which are no Markov processes are given.