Recurrence and invariant measure of Markov chains in double-infinite random environments

The concepts of π -irreduciblity, recurrence and transience are introduced into the research field of Markov chains in random environments. That a π -irreducible chain must be either recurrent or transient is proved, a criterion is shown for recurrent Markov chains in double-infinite random environments, the existence of invariant measure of π -irreducible chains in double-infinite environments is discussed, and then Orey’s open-questions are partially answered.