Chains and Specifications
2004, v.10, Issue 3, 435-456
We review four types of results combining or relating the theories of discrete-time stochastic processes and of one-dimensional specifications. First we list some general properties of stochastic processes which are extremal among those consistent with a given set of transition probabilities. They include: triviality on the tail field, short-range correlations, realization via infinite-volume limits and ergodicity. Second we detail two new uniqueness criteria for stochastic processes and discuss corresponding mixing bounds. These criteria are analogous to those obtained by Dobrushin and Georgii for Gibbs measures. Third, we discuss conditions for a stochastic process to define a Gibbs measure and vice versa, that generalize well known equivalence results between ergodic Markov chains and fields. Finally we state a (re)construction theorem for specifications starting from single-site conditioning, which applies in a rather general setting.
Keywords: discrete-time stochastic processes,Gibbs measure,chains with complete connections,Markov chains,ergodicity and rates of mixing