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

TL;DR

This paper introduces the concept of dynamic uniqueness for $g$-measures, establishing its equivalence to a weak-$ { ext{l}}^2$ summability condition, and demonstrates its implications for mixing properties and stability of stochastic chains.

Contribution

It defines dynamic uniqueness, proves its equivalence to a weak-$ { ext{l}}^2$ condition, and links this to mixing properties of $g$-measures, strengthening existing criteria.

Findings

Dynamic uniqueness is stronger than usual uniqueness.

Weak-$ { ext{l}}^2$ condition characterizes dynamic stability.

Weak-$ { ext{l}}^2$ criterion implies $eta$-mixing.

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-\"Oberg $\ell^2$ 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.