The mixing properties of certain processes related to Markov chains

Summary

Let {Y _n ,− ∞ < n < ∞} be either a function of a stationary Markov chain with countable state space, or a finitary process in the sense of Heller [3]. The purpose of this note is to prove that if {Y _n ,− ∞ < n < ∞} is mixing, then it is aK-shift. (Definitions will be given below.)

IfT is a measure-preserving transformation on a probability space, then the following implications relevant to the present paper are known: (1)T is aK-shift ⇒T is (r + 1)-mixing ⇒T isr-mixing ⇒T is totally ergodic ⇒T is ergodic, and (2)T is ergodic ⇏T is totally ergodic ⇏T isr-mixing ⇏T is aK-shift.

It is not known if the classes ofr-mixing and (r + 1)-mixing transformations are distinct. (1-mixing is also called mixing.) The results of this note then imply that for the classes of transformation that we are consideringr-mixing and (r + 1)-mixing are equivalent.