Uniform ergodicity and strong mixing

as n---,oe. It should be noted that what we call strong mixing in this paper is a more stringent condition than what is ordinarily referred to as strong mixing, namely that P(A c~ zkB) ~P(A) P(B) as k ~ o o for any two measurable sets A, B. It is clear that strong mixing implies uniform ergodicity. In the case of stationary Markov sequences these conditions can be stated in another convenient form. If {X,} is a Markov sequence let d be the Borel field of measurable sets on the state space of the process. Let P ( , . ) be the one-step transition probability function of the Markov sequence and set