The tail $\sigma$-field of a finitely additive Markov chain starting from a recurrent state

Abstract:

For a Markov chain with an arbitrary nonempty state space, with stationary finitely additive transition probabilities and with initial distribution concentrated on a recurrent state, it is shown that the probability of every tail set is either zero or one. This generalizes and in particular gives an alternative proof of the result due to Blackwell and Freedman [1] in case the state space is countable and all transition probabilities are countably additive.