Can a mixing process be non-stationary?

Question

I was always under the impression that a mixing process is ergodic and an ergodic process is necessarily stationary, so that a mixing process is stationary. I have come across a paper discussing non-stationary $\phi$-mixing processes, and I wonder if those even exist. Many thanks!

Answer 1

Consider the case of an i.i.d. sequence $(\varepsilon_j)_{j\in\mathbb Z}$ and define $X_j := j\varepsilon_j$. The sequence $\left(X_j\right)_{j\in\mathbb Z}$ is independent hence $\phi$-mixing but not stationary, even in the wide sense.

Answer 2

I think here's where I was confused:

A transformation $T$ is mixing if $\lim_{n\rightarrow \infty} P(A \cap T^{-n}B)=P(A)P(B)$ Now $T$ need not be measure-preserving, but if it is, then its mixing property will imply that it's ergodic since then we'll have $P(A \cap T^{-n}B)=P(A \cap B) = P(A)P(B)$ where the first equality follows from the fact that $T$ preserves $B$ and the second equality is due to the mixing assumption. Now if we let $A=B$ we get $P(B)=P^2(B)$ so that $P(B) = 0$ or $P(B)=1$ for all invariant sets, hence $T$ is ergodic.

So stationarity along with mixing together imply ergodicity, but mixing by itself doesn't necessarily imply ergodicity.