The Weak Irreducibility Markov Chains

Abstract

Irreducibility serves as the foundational concept in the theoretical study of Markov chains. This paper introduces a generalized notion termed weak irreducibility and investigates its stochastic stability properties for Markov chains defined on countable state spaces. We establish the existence of invariant measures and stationary distributions for weak irreducibility Markov chains and provide an illustrative example demonstrating its practical relevance. Moreover, we extend the framework of weak irreducibility to general state spaces without requiring the prior assumptions of separability and the existence of a reference measure.