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 , for large n, with probability near one, the trajectories of the MC are in states i, where is either near 0 or near 1. A similar statement holds for the entrance σ-algebra when n tends to . To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by or 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.
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