Talk:Ergodic theory
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[edit] Definition
Hola. I've stubbed this page with a defn of "ergodic" and a hand-wavy statement of the ergodic theorem. There are links to ergodic theory and ergodic hypothesis. I don't know how we want to split things up. Should ergodic, ergodic theorem, and ergodic theory all get separate pages? Put 'em all on the same page? I guess at this point my inclination is to put them all together since there is little material so far. Happy editing, Wile E. Heresiarch 02:30, 19 Feb 2004 (UTC)
I think this page should be moved to ergodic theory. I don't like using an adjective as a page title. This should be a redirect page. Michael Hardy 22:33, 13 Mar 2004 (UTC)
- I agree about the page title. I will put in a request for the ergodic theory redirect to be removed so that the ergodic page can move there. Wile E. Heresiarch 18:03, 15 Mar 2004 (UTC)
It would be really nice if someone could come up with a sentence to put in the first paragraph that would make sense to a typical undergraduate math major. (I'm not even gonna say "typical reader"!) Maybe that's just not possible given the topic? - dcljr 03:45, 20 Jul 2004 (UTC)
Hi, I went here to find something on ergodicity as in "ergodic Markov model", more precisely the stuff with stationary distributions -- on one hand, it takes a little thinking to match the abstract stuff here with the very constrained notions used for MMs (if you're vaguely familiar with them), on the other, the page on MMs just handwavingly refers to here. I think that MMs would be a fine illustrating example here, but I'm not familiar enough with ergodic theory to write up (or even think up) a good formulation - Yannick V.
[edit] Revised intro
My proposal for a more amenable and general statement:
Generally, an ergodic theorem refers to any statement about the existence of a mean value with respect to trajectories of a random process taken with respect to time. Intuitively it means that the mean of a random process is irrespective of it's starting point.
A subsection with ergodic theorem should be included.
- Sounds good to me. linas 00:36, 16 December 2005 (UTC)
This is a somewhat pedantic point, but strictly speaking the equidistribution theorem is not a special case of the ergodic theorem: the equidistribution theorem gives pointwise convergence at every point, which is a stronger statement than the Lebesgue-a.e. convergence given by applying the Birkhoff theorem. 193.170.117.12 14:35, 10 May 2006 (UTC)
As an electrical engineer graduate looking for info concerning stationarity and ergodicity in regards to statistics, I think this page is way to mathematical. I'm (and probably a lot of other students studying signal processing) looking for how to distinguish between stationarity and ergodicity where ergodicity is considered a more sringent requirement. At least a section ought to be included on ergodicity in regards to random processes. —Preceding unsigned comment added by Perman07 (talk • contribs) 23:50, 28 January 2008 (UTC)
[edit] copyright violation?
The first two paragraphs copy verbatim the 3rd and 4th paragraphs of the linked article at http://news.softpedia.com/news/What-is-ergodicity-15686.shtml. Of course, the copy could have gone the other direction. It's not clear if this is small enough to be fair use, but it should certainly be credited.
- Those paragraphs were added by OO0000OO on 26 March 2006. As of 11 April 2006 those paragraph are the user's sole contribution to the WP. I have removed the section, as Softpedia claims the material copyrighted. XaosBits 21:38, 11 April 2006 (UTC)
I'm sorry to be a pill. I stumbled into this page and was very interested in what it might say. I get the feeling that the current page is technically sound. I don't think it is sound in terms of explaining why anyone would care about ergodic theory. What problems does it arise in? Why was it thought to be of sufficient interest that it was pursued? What are its uses now that it is known? How would you know if you stumbled across a problem that needs it? etc.
Again, I'm really sorry to be picky. I think the material here is probably quite good.
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- Three weeks ago, I stumbled across this page trying to solve a cute little problem suggested to me by my masters student. The statement of this problem can be understood by any high school student, even a bright 7th grader, but that same high school student would not be able to work it out without using (or re-proving) the ergodic theorem. I will post that problem on the page, if there's any interest. Vegasprof 00:42, 20 May 2007 (UTC)
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- Please, do so. Jayme 21:37, 9 August 2007 (UTC)
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[edit] Nutrition
In the field of nutrition, "transit time" is the time it takes for (indigestible) food to pass from the mouth completely through the body. When I used Wikipedia's search box to seek it, I was expecting a page about nutrition, but was redirected here. D021317c 23:04, 11 November 2007 (UTC)
[edit] Vague or Innacurate Statement of Mean Ergodic Theorem?
The final phrase in the statement of the Mean Ergodic Theorem ("... where the limit is in the L^2 sense") seems vague or innacurate to me. The phrase describes a sequence of operators U^n converging *pointwise* to an operator P. To be sure, the pointwise convergence appears to be in the topology induced from the inner product on the Hilbert space. The phrase "in the L^2 sense" is consistent with that convergence, I suppose, but also sounds consistent with other typical topologies (norm, strong, weak, weak*) that might also involve L^2.
Unfortunately, I don't know the precise statement of the theorem to correct it. 65.211.34.130 (talk) 17:13, 21 April 2008 (UTC)