One dimensional Markov random fields, Markov chains and Topological Markov fields

Details One dimensional Markov random fields, Markov chains and Topological Markov fields One dimensional Markov random fields, Markov chains and Topological Markov fields

2011-12-19

arxiv.org/abs/1112.4240v1

Mathematics

Dynamical Systems

15 pages

Scientific paper

In this paper we show that any one-dimensional stationary, finite-valued Markov Random Field (MRF) is a Markov chain, without any mixing condition or condition on the support. Our proof makes use of two properties of the support $X$ of a finite-valued stationary MRF: 1) $X$ is non-wandering (this is a property of the support of any finite-valued stationary process) and 2) $X$ is a topological Markov field (TMF). The latter is a new property that sits in between the classes of shifts of finite type and sofic shifts, which are well-known objects of study in symbolic dynamics. Here, we develop the TMF property in one dimension, and we will develop this property in higher dimensions in a future paper. While we are mainly interested in discrete-time finite-valued stationary MRF's, we also consider continuous-time, finite-valued stationary MRF's, and show that these are (continuous-time) Markov chains as well.

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