A Zero-One Law for Markov Chains

Abstract

We prove an analog of the classical Zero-One Law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event $A$ from the tail $\sigma$-algebra of MC $(Z_n)$, for large $n$, with probability near one, the trajectories of the MC are in states $i$, where $P(A|Z_n=i)$ is either near $0$ or near $1$. A similar statement holds for the entrance $\sigma$-algebra, when $n$ tends to $-\infty$. To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by ${\mathbb{Z}}_{-}$ or $\mathbb{Z}$ in both the finite and countable cases. This extends a well-known result due to Kolmogorov. Further, in our discussion, we note an interesting dichotomy between two commonly used definitions of MCs.