Relations Between Ergodicity and Mean Drift for Markov Chains¹

If {X n } is an irreducible aperiodic Markov chain on a state apace denote the mean one step change of position, or “drift”, of {X n } at j. The main result of this note is to show that, when |µ(j)| is bounded, {X n } admits a stationary distribution {π j }if and only if for some N > 0 and some state i, lim inf ∑ when this holds, the limit infimum is in fact . Many of the known sufficient or necessary criteria for the existence of a stationary distribution can then be derived easily from this and related results.