# Different definitions of ergodicity for stationary processes

From page 3 of a note:

A stationary process is ergodic if any two variables positioned far apart in the sequence are almost independently distributed.

A formal definition is the following: $\{w_t\}$ is ergodic if for any two bounded functions $f$ in $(k + 1)$ variables and $g$ in $(l + 1)$ variables: $$\lim_{N \to \infty} |E(f(w_t, ..., w_{t+k})g(W_{t+N}, ..., w_{t+N+l}))|- |E(f(w_t, ..., w_{t+k}))| |E(g(W_{t+N}, ..., w_{t+N+l}))| =0.$$

This definition of ergodicity can also apply to wide-sense stationary processes, does it?

This definition of ergodicity seems more similar to mixing than to ergodicity. So is it actually a type of mixing for (wide-sense) stationary processes?

Is this definition of ergodicity for a stationary process, same as the mean-square ergodicity in the first and second moment for wide-sense stationary processes, or ergodicity for a measure-preserving mapping?

Thanks and regards!

• The condition you have given is, as you say, a mixing condition and not an ergodicity condition. For aperiodic Markov chains, the two notions coincide. For non-aperiodic MCs, they don't. – Anthony Quas Jan 30 '14 at 19:25
• Thanks, @AnthonyQuas! Could you take a look at a reply? What does the author mean by "mixing" in the last sentence of the first paragraph ( The notion of ergodicity you are referring to would rather correspond to what is called "mixing" in ergodic theory)? What he refers to is "ergodicity" in Markov chain, which seems to me more like a stronger version of irreducibility, rather than "mixing" in ergodicity theory of dynamic system. What do you think about my thought? – StackExchange for All Jan 30 '14 at 19:43
• Mixing in ergodic theory is the condition that $\mathbb P(A\cap T^{-n}B)\to \mathbb P(A)\mathbb P(B)$, essentially exactly the same as your condition. On the other hand, ergodicity is that condition that $\lim_{N\to\infty}(1/N)\sum_{n=1}^N \mathbb P(A\cap T^{-n}B)\to \mathbb P(A)\mathbb P(B)$. As I said in my earlier comment irreducible finite state Markov Chains that are not aperiodic are ergodic but not mixing. – Anthony Quas Jan 30 '14 at 22:28
• @AnthonyQuas: Thanks! The questions in my last comment wasn't clear, and they are about a reply here mathoverflow.net/a/74505/5142. What does the author mean by "mixing" in the last sentence of the first paragraph: "The notion of ergodicity you are referring to would rather correspond to what is called 'mixing' in ergodic theory"? What he refers to is "ergodicity" in Markov chain, which seems to me more like a stronger version of irreducibility, rather than "mixing" in ergodicity theory of dynamic system. What do you think about my thought? – StackExchange for All Jan 30 '14 at 22:43
• @AnthonyQuas: The definition of ergodicity in your comment is called weak mixing in Wikipedia (en.wikipedia.org/wiki/Mixing_%28mathematics%29). What I have in mind about ergodicity from reading Wikipedia is that starting anywhere, you can go everywhere. What I have in mind about mixing from reading Wikipedia is that starting anywhere, after a sufficiently long time, where you will be is "independent" from where you start. Are my understandings correct? – StackExchange for All Jan 30 '14 at 22:49