Characterization of the stability of chains associated with g-measures

الأصل · EN

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 σ-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-ℓ² summability condition on the kernel. This generalizes and strengthens the Johansson-Öberg ℓ² criterion for uniqueness of g-measures. Finally, among other things, we prove that the weak-ℓ² criterion implies β-mixing of the unique g-measure compatible with a regular kernel improving several results in the literature.