One dimensional Markov random fields, Markov chains and Topological Markov fields
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2011
- DOI:
- 10.48550/arXiv.1112.4240
- arXiv:
- arXiv:1112.4240
- Bibcode:
- 2011arXiv1112.4240C
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Probability;
- 37B10;
- 60J10;
- 60J27
- E-Print:
- 15 pages