Characterization of the stability of chains associated with $g$-measures

Abstract

In this paper we introduce a notion of asymptotic stability of a probability kernel, which we call dynamic uniqueness. We say that a kernel exhibits dynamic uniqueness if all the stochastic chains starting from a fixed past coincide on the future tail $\sigma$-algebra. We prove that the dynamic uniqueness is generally stronger than the usual notion of uniqueness for $g$-measures. Our main result shows that dynamic uniqueness is equivalent to the weak-${\ell }^2$ summability condition on the kernel. This generalizes and strengthens the Johansson-Öberg ${\ell }^2$ criterion for uniqueness of $g$-measures. Finally, among other things, we prove that the weak-${\ell }^2$ criterion implies $\beta$-mixing of the unique $g$-measure compatible with a regular kernel improving several results in the literature.

We did a considerable editing of the manuscript. We wanted to emphasize more the weak-${\ell }^2$ criterion. We modified the tile, abstract and intro

References in corpus (2)

• Unique Bernoulli g-measures
• Attractive regular stochastic chains: perfect simulation and phase transition