I've already asked this question on [math.stack](https://math.stackexchange.com/questions/834312/strange-definition-of-ergodicity) 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!



Strange definition of ergodicity
Yes. That is the same thing as ergodicity. To explain it, you have probably seen somewhere that the way to understand random variables formally is functions from a (hidden) underlying space $\Omega$ to $\mathbb R$. That is: knowing the point $\omega$ of $\Omega$, one can recover $X\_n(\omega)$ for each $n\in \mathbb Z$. 


In fact, a standard $\Omega$ to use is just the space of sequences of real numbers. Then the function $X\_n$ is just the $n$th coordinate function. If you do this, then $\Omega$ is equipped with a natural map, namely the shift map, $\sigma$. You can now define a version of your function $g$ on $\Omega$, namely $G(\omega)=g(\omega\_0,\ldots,\omega\_k)$. The left side of your equality is now $\lim\_{N\to\infty}(1/N)\sum\_{n=1}^N G(\sigma^n\omega)$. 
The right side is $\int G(\omega)\,d\mu(\omega)$, where $\mu$ is the distribution on the set of sequences occurring. Notice: invariance of $\mu$ corresponds exactly to stationarity of the sequence of random variables. 


