Attractive regular stochastic chains: perfect simulation and phase transition

Abstract

We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an i.i.d. process with countable alphabet, (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.

22 pages, 1 pseudo-algorithm, 1 figure. Minor changes in the presentation. Lemma 6 has been removed

References in corpus (3)

• Variable length Markov chains and dynamical sources
• A general framework for perfect simulation of long memory processes
• Perfect simulation for locally continuous chains of infinite order

Cited by in corpus (7)

• On concentration inequalities and their applications for Gibbs measures in lattice systems
• One-sided continuity properties for the Schonmann projection
• Optimal Gaussian concentration bounds for stochastic chains of unbounded memory
• Gaussian Concentration bound for potentials satisfying Walters condition with subexponential continuity rates
• Stochastic processes with random contexts: a characterization, and adaptive estimators for the transition probabilities
• Explicit estimates in the Bramson-Kalikow model
• Characterization of the stability of chains associated with $g$-measures