Asymptotically mean stationary sources

Inverse images of product cylinders under $T$ are product cylinders; since these generate the product sigma-algebra, $T$ is measurable. The law $P$ is asymptotically mean stationary (AMS) when for every measurable event $E$ the limit $\bar{P}(E)={\lim }_{N\to \mathrm{\infty }}\frac{1}{N}{\sum }_{j=0}^{N-1}P(T^{-j}E)$ exists and the resulting set function is a probability measure.

For each measurable $E$, the Ces`aro averages for $T^{-1}E$ and $E$ differ by $\frac{1}{N}{\sum }_{j=0}^{N-1}P(T^{-(j+1)}E)-\frac{1}{N}{\sum }_{j=0}^{N-1}P(T^{-j}E)=\frac{P(T^{-N}E)-P(E)}{N}\mathrm{.}$ The numerator has absolute value at most $1$, so its quotient tends to zero. Taking limits, which exist by step 1.1 for both events, gives $\bar{P}(T^{-1}E)=\bar{P}(E)$. Thus the Ces`aro limit is stationary in the sense of [F1].

The resulting law $\bar{P}$ is called the stationary mean when it is stationary in the sense of [F1]. If $E$ is strictly shift invariant, then each summand $P(T^{-j}E)=P(E)$; hence $P(E)=\bar{P}(E)$. Tail events are handled by the separate domination lemma, not by this invariant-event identity.

Call the AMS source ergodic when its stationary mean $\bar{P}$ is ergodic as a measure-preserving system under the same shift. This is a property of the mean, not a claim that $P$ itself is stationary.