Uniqueness of stationary compatible probability measures for chains of infinite order with forbidden transitions

TL;DR

This paper establishes conditions for the uniqueness of stationary measures in infinite-order chains with forbidden transitions, extending previous criteria and illustrating with concrete examples.

Contribution

It extends the $ ext{ell}^2$-based uniqueness criterion to chains with forbidden transitions, broadening applicability beyond strongly non-null chains.

Findings

Provided sufficient conditions for uniqueness of stationary measures.

Extended Johansson and Öberg's $ ext{ell}^2$ criterion to new classes of chains.

Illustrated results with concrete examples and literature comparison.

Abstract

In this paper, we consider chains of infinite order on countable state spaces with prohibited transitions. We give a set of sufficient conditions on the structure of the probability kernels of the chains to have at most one stationary probability measure compatible with the kernel. Our main result extends the uniqueness $\ell^2$ criterion from Johansson and \"Oberg (2003) which was obtained for strongly non-null chains. A particular attention is given to concrete examples, illustrating the main theorem and its corollaries, with comparison to results of the existing literature.