Timeline for Different definitions of ergodicity for stationary processes
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| Jan 31, 2014 at 9:43 | comment | added | R W | @Tim No - it is not - please read carefully the end of the section "Mixing in dynamical systems" in the wiki article you are referring to - there they talk about differences between strong mixing, weak mixing and ergodicity. Although wikipedia is right in this concrete case, the problem with most so called "popular" expositions is that they rather confuse the reader - so it's better to try to rely on more serious sources. | |
| Jan 30, 2014 at 22:49 | comment | added | Tim | @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? | |
| Jan 30, 2014 at 22:43 | comment | added | Tim | @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? | |
| Jan 30, 2014 at 22:28 | comment | added | Anthony Quas | 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. | |
| Jan 30, 2014 at 19:43 | comment | added | Tim | 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? | |
| Jan 30, 2014 at 19:25 | comment | added | Anthony Quas | 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. | |
| Jan 30, 2014 at 17:46 | history | asked | Tim | CC BY-SA 3.0 |