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 than ergodicity in the sense of the Birkhoff ergodic theorem. Here the boundedness of $g$ means the property would be equivalent only for indicator $g$'s (because of DCT I think). On the other hand, any $k$-tuple of $X_i$'s is allowed.. Also, is there an insightful bit of intuition for this property as there are for normal ergodicity, and mixing?

Thanks in advance!

  • 3
    $\begingroup$ Take your sample space to be sequence space that you get for the joint law of all of the $X_i$ by using Kolmogorov's extension theorem and let your measure preserving transformation be the shift map taking you from coordinate $i$ to coordinate $i+1$. I think that this condition says that that shift map is ergodic in the usual definition. Proving that requires a bit of measure theory, since you need to show that the indicator of any invariant set is well approximated by a function of $k$ variables in the right sense. There may be some technical issues that make this a bit weaker though. $\endgroup$ Commented Jun 14, 2014 at 21:24
  • $\begingroup$ Hi, thanks for your comment! I couldn't understand it and decided to wait a few days for an answer. Eventually I asked on math.overflow and received an answer very similar to yours. I'll gladly accept your answer if you write one! $\endgroup$
    – user153312
    Commented Jun 19, 2014 at 7:32


You must log in to answer this question.