A zero-one law for Markov chains

Abstract

We prove an analogue 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 σ-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 σ-algebra when n tends to $-\mathrm{\infty }$. 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.

Keywords:

• Nonhomogeneous Markov chains
• zero-one laws
• tail sigma-algebra
• entrance sigma algebra

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of M. Grabchak was supported in part by Russian Science Foundation Project No. 17-11-01098.