I've already asked this question on math.stack a few days ago and haven't received an answer, so I'm asking here.

In an engineering course, a stationary process was defined to be ergodic if for all $k\in \mathbb{N}$ and for any bounded (measurable) function of $k+1$ variables we have $$\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n=1}^N g(X_n,\dots,X_{n+k})\overset{\text{a.s}}{=}Eg(X_n,\dots,X_{n+k})$$ From the little I've read about ergodic theory, this does not seem familiar nor does it seem to fit into the ergodic hierarchy I know, i.e ergodic, weak mixing, strong mixing etc. It seems like a different property from ergodicity (in the sense of the Birkhoff ergodic theorem). Here the boundedness of $g$ means a formulation with indicator $g$'s would be equivalent (because of DCT I think). On the other hand, any $k$-tuple of $X_i$'s is allowed.. Is there an insightful bit of intuition for this property as there are for normal ergodicity, and mixing? Where does it fit into the ergodic hierarchy?

Thanks in advance!