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arXiv · 1112.4240

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

Nishant Chandgotia·Guangyue Han·Brian Marcus·Tom Meyerovitch·Ronnie Pavlov
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Abstract

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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Nishant Chandgotia, Guangyue Han, Brian Marcus, Tom Meyerovitch, Ronnie Pavlov. 2011-12-19. One dimensional Markov random fields, Markov chains and Topological Markov fields. https://doi.org/10.1090/s0002-9939-2013-11741-7

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