A Zero-One Law for Markov Chains

TL;DR

This paper establishes a zero-one law for both homogeneous and nonhomogeneous Markov chains, showing that tail and entrance events become almost surely trivial as the chain evolves, extending Kolmogorov's classical result.

Contribution

It extends the classical zero-one law to nonhomogeneous Markov chains and provides detailed conditions for their existence on countable and finite state spaces.

Findings

Zero-one law applies to tail and entrance events in Markov chains.

Extension of Kolmogorov's result to nonhomogeneous chains.

Detailed existence conditions for nonhomogeneous chains on various index sets.

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.