# Talk:Ergodic theory

## 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.

I came here looking for a quick reminder what ergodicity means (which is redirected here). Needless to say, this article is not very helpful in that respect. I think we need a kind of article/disambiguation page for the word ergodic and its variants that explains quickly the idea that the word represents, and then links to pages like these. 145.18.110.208 (talk) 11:48, 14 August 2008 (UTC)

I guess we already have one. I've changed the redirect for ergodicity. 145.18.110.208 (talk) 11:52, 14 August 2008 (UTC)

## 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 (talkcontribs) 23:50, 28 January 2008 (UTC)

I agree with the poster above. This page is perfectly opaque to someone without a background in mathematics. Try using a real world example, or a metaphor, or anything! —Preceding unsigned comment added by 74.85.239.73 (talk) 07:34, 22 April 2009 (UTC)

## a good example need

The concept of ergodicity is entering several fields, such as ecology, where not everyone is comfortable with the mathematics level of this article. For example, recently a friend asked me what ergodicity meant, and the first think I thought of was to refer her to the wikipedia. But when I went to check this article I realized it would have been incomprehensible to her. So I do think a simple, without appealing to mathematical definitions, would be most useful for a general audience.

I will try to give a simple explanation. If you have a record of a random process it is natural to calculate statistical parameters (mean, variance, auto-correlation) using just this one sample of the random process. But then there is the question of how typical of the process is your record. If you had worked using a different record would you have got the same results to within statistical error? The assumption that you are justified in using your one record is called the ergodic assumption. It is very often mentioned in journal papers although it is never stated how you know whether the process is ergodic so you are back to square one. Books usually omit to say that certain processes have actually been proved ergodic, the most important examples being the stationary Gaussian process having a continuous spectrum and more generally processes derived by filtering white noise (Gaussian or shot noise). Also Markov processes, both in discontinuous time and continuous time, are ergodic if you can go from any state to any other state so the process does not break up into pieces (Markov's theorem).JFB80 (talk) 20:22, 24 April 2011 (UTC)JFB80 (talk) 11:06, 26 April 2011 (UTC)
OK that's getting closer to english I can understand. Are you saying that the 'ergodic assumption' is that if you and I go out and each take a sample of the same thing, our samples will give the same statistical results (give or take the odd anomaly)? and that this has been proved for some things that are 'ergodic' and that presumably there are some things that are proved to be 'unergodic' and other things which are mostly "assumed to be ergodic"? EdwardLane (talk) 16:12, 21 March 2012 (UTC)
Yes roughly if your 'thing' is a random process which is the same for both of you. In practice random processes are usually "assumed to be ergodic".JFB80 (talk) 20:06, 22 March 2012 (UTC)
OK so I think this might need clarification that it has to be a random process? And not a random sample of people as to whether they voted A or B?EdwardLane (talk) 13:23, 23 March 2012 (UTC)
Ergodicity only has meaning for random processes. It is usually stated 'time-average=ensemble average'. There must be a time average. In your case you dont need it.JFB80 (talk) 19:23, 23 March 2012 (UTC)
OK, so I've still not got my head around what this ergodicity 'is' I'll have another go at smashing my way through the article and see if I can understand it. Once I get there I might be able to find some wording that explains it for the interested layman.EdwardLane (talk) 10:34, 26 March 2012 (UTC)

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.

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)
Please, do so. Jayme 21:37, 9 August 2007 (UTC)

## 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)

## 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)

## connections with other areas of mathematics

"Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions)."

It would be helpful if someone summarized these connections or provided links about them 87.203.88.226 (talk) 15:30, 10 October 2008 (UTC)

## Article merge

IMHO it would be nice to merge Ergodic_(adjective) into this article, since the topic of the two pages is the same, and it has almost nothing in extra compared to this page. However, I'd like to let this job for someone with more expertise in the field than myself. AdamSiska (talk) 01:07, 29 January 2009 (UTC)

Having both separate articles might potentially be useful at some later point in time. This is not unprecedented: for instance, we have both Group (mathematics) to refer to the general notion, and Group theory to refer to the specialized field of study. That said, as things currently stand I think that almost all of the current content at Ergodic (adjective) may be better suited to this article. Sławomir Biały (talk) 16:23, 26 August 2009 (UTC)
I believe the adjective has an interesting history in the development of statistical physics prior to the existence of what is now called Ergodic Theory. (Specifically, that statistical physics developers made a hidden assumption in their arguments that became known as the "ergodic assumption.") The history of this developing concept would not belong under the heading of Ergodic theory because in Ergodic theory the term is no longer developing; it has a precise mathematical definition. I would provide more information but I am not an expert-- I'm just saying there

is reason to keep the pages separate. —Preceding unsigned comment added by 4.243.98.5 (talk) 20:00, 9 November 2009 (UTC)

Take a look to other uses of the term Ergodic, for instance Ergodic Literature. Pablo Martínez Merino. 14:25, 31 August 2009 (UTC)
Pablo, to what end? PDBailey (talk) 13:48, 20 October 2009 (UTC)

Moved (modded) definition to ergodic (adjective). No longer a need for Ergodic theory#Ergodic transformations or a merger. We can move the examples to an independent section. Create an overview section to lighten the intro. Tiled (talk) 01:02, 12 September 2010 (UTC)

## Ergodic splitting in Lp

I'm thinking that it would be nice to add to the article a brief section on the ergodic decomposition given by the von Neumann's theorem:

${\displaystyle L^{2}(X,\Sigma ,\mu )=L^{2}(X,\Sigma _{T},\mu )\oplus {\overline {\{f-f\circ T\,:\,f\in L^{2}(X,\Sigma ,\mu )\}}},}$

that occurs for a measure-preserving map ${\displaystyle T}$ of a probability space ${\displaystyle (X,\Sigma ,\mu )}$, ${\displaystyle \Sigma _{T}}$ being the sub-σ-algebra of all ${\displaystyle T}$-invariant measurable sets, and the (orthogonal) projector being the conditional expectation ${\displaystyle E(\cdot |\Sigma _{T})}$. For all ${\displaystyle 1\leq p\leq \infty }$, the conditional expectation is well-defined as a linear projector of norm 1 on ${\displaystyle \textstyle L^{p}(X,\Sigma ,\mu )}$, with range the closed subspace ${\displaystyle L^{p}(X,\Sigma _{T},\mu )\subset L^{p}(X,\Sigma ,\mu )}$. Therefore it's quite natural to consider the analogue decomposition of the ${\displaystyle L^{p}}$ spaces given by the${\displaystyle L^{p}}$ projector , that should be:

${\displaystyle L^{p}(X,\Sigma ,\mu )=L^{p}(X,\Sigma _{T},\mu )\oplus {\overline {\{f-f\circ T\,:\,f\in L^{p}(X,\Sigma ,\mu )\}}}^{L^{p}}.}$

Now if ${\displaystyle 1\leq p\leq 2}$, this splitting actually holds true and is easily obtained with a Lp-closure starting from the ${\displaystyle L^{2}}$ splitting. For ${\displaystyle 2\leq p\leq \infty }$, the splitting is obtained by restriction to ${\displaystyle L^{p}(X,\Sigma ,\mu )}$. And here comes the problem: this way one gets the right first factor (the range of the projector) ${\displaystyle L^{p}(X,\Sigma _{T},\mu )=L^{2}(X,\Sigma _{T},\mu )\cap L^{p}(X,\Sigma ,\mu )}$, but I can't see why the kernel of the projector,

${\displaystyle {\overline {\{f-f\circ T\,:\,f\in L^{2}(X,\Sigma ,\mu )\}}}^{L^{2}}\cap L^{p}(X,\Sigma ,\mu )}$

should be equal to (and not larger than)

${\displaystyle {\overline {\{f-f\circ T\,:\,f\in L^{p}(X,\Sigma ,\mu )\}}}^{L^{p}}.}$

It's not a fundamental point (it does not enter in the proof of the main ergodic theorems) but I think that if a complete analogy holds true it would be nice to state it, and if it doesn't, one would like to know what goes wrong. I checked the main texts of ergodic theory on this point, and found nothing. I posted this question to the RefDesk too. --pma (talk) 10:16, 20 October 2009 (UTC)

## Birkhoff's 'Ergodic Theorem'

Birkhoff's famous 'ergodic theorem' was misnamed - it was NOT a proof of the ergodic theorem but only a proof of the existence of time averages. Then another assumption called metric transitivity (here renamed ergodicity of the transformation) was introduced to deduce ergodicity (that assumption being equivalent to ergodicity of course). So as stated, the article makes no sense: In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic theorem, ..etc. i.e. if we assume it is ergodic then it is ergodic. Brilliant! JFB80 (talk) 21:03, 24 April 2011 (UTC)

Your point is not valid. In fact, Khintchine answers objections such as yours in his book. In the abstract, your objection would apply to every logical equivalence: «This theorem has no content since, after all, the conclusions follow from the premisses».

You say 'your point is not valid' without saying why not. You quote Khinchine but don’t say which book it is, or where the answer to the objection is to be found. So Citation needed Then you make some vague general assertion about mathematical proofs. These are just red herrings. You cannot deny the assertion that Birkhoff's theorem does not prove the ergodic property but only the existence of a time average and and so is misnamed Birkhoff's 'ergodic' theorem. JFB80 (talk) 10:43, 9 September 2012 (UTC)

## Why Birkhoff--Khinchin?

Why does Khinchin get his name on this? This usage is non-standard and unjustified, as far as I know, but if you have some reason, here is your chance to say it..... — Preceding unsigned comment added by 173.70.2.241 (talk) 15:41, 30 July 2012 (UTC)

I believe you are the same person 173.70.2.241 who wrote in the last section. You appear to have been able to open a new section while remaining anonymous in the previous section. Now you ask ' why does Khinchin get his name on this'. Judging from your previous comment you should know. Actually the answer to your question might be found in the article if you read it carefully. You say the inclusion of Khinchin is unjustified. How can you know if you don’t know the reason why the name was included ?JFB80 (talk) 10:46, 9 September 2012 (UTC)