Talk:Stationary process

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No Wikipedia article should begin by saying "A stationary process is one in which...". Some context setting is needed first, at least by saying which discipline is being discussed; e.g., is this about law, about software, etc.? The first sentence seems confused to me. But I'm not sure what the person who wrote it had in mind. Michael Hardy 01:50, 29 Sep 2004 (UTC)

... maybe a stochastic process with stationary increments?? Michael Hardy 01:51, 29 Sep 2004 (UTC)

I believe the {{technical}}, which has been up for two years, is outdated. The comments by Michael Hardy have already been corrected. --Zvika 15:09, 13 October 2006 (UTC)

[edit] stationary signal

"in signal processing, a stationary signal is a signal whose frequency content does not change over time." Is this true? Is it the same thing? - Omegatron 23:29, Sep 29, 2004 (UTC)

[edit] Stationary process in statistics

(EDIT - DELETED NONSENSE) Alternative examples of non-stationary processes are stock prices, economic aggregates (e.g. GDP) and the position of a gas particle in space. In all of these examples the long run distribution of the level/position depends on the current level/position violating the constancy of the (EDIT) unconditional (EDIT) distribution required for stationarity.

there are many ways to extend this article with brief discusison of: history, applications, differencing to induce stationarity (orders of non-stationarity), linear systems (ARMA processes, VARs, GARCH processes etc), difference between trending and non-stationary series, "unit roots", spurious regressioon, link to cointegration, link to brownian motion etc. --Cripes 21:28, 27 March 2006 (UTC)

In books on stochastic processes and on time series, I have seen stationarity defined as meaning the probability distribution of X(t) is the same for all values of t. And you seem to contradict yourself concerning the cymbal: if it decays, then its probability distribution approaches a point mass as t approaches ∞; therefore it is stationary according to the definition you gave us. Michael Hardy 21:38, 27 March 2006 (UTC)

you're right about the definition, apologies - i've edited out the nonsensical parts of my previous post. I was mis-interpreting your statement of "distribution" as the "conditional distribution" (which is not necessarily constant). since i am finding it necessary at work to brush up on this stuff i might offer some additional content for the page (as per the list) if you are interested in editing it --Cripes 22:40, 27 March 2006 (UTC)

[edit] Stationarity implies WSS?

Please try to work things out in the talk page rather than reverting each other's edits. IMO, User:128.214.205.6 is technically correct: A process cannot be said to be WSS if its mean or variance do not exist, but it could still be stationary (e.g. an iid process where each time sample is Cauchy distributed). However, I think this is something of a pathological case; in most cases of interest, the moments exist and so stationary implies WSS. So perhaps User:128.214.205.6's sentence is more confusing than instructive. I tried to change the wording, I hope you will like it. --Zvika 19:10, 16 October 2006 (UTC)

This was my first revision of the material in Wikipedia. Hence please excuse my ignorance and reverting "each other's edits". I hope editing this talk page is the proper method to discuss revisions. (Is it?) Now to the topic itself: The above mentioned iid Cauchy series is a well-known theoretical case in point. Strictly stationary while not weakly stationary processes can arise while trying to design empirically relevant processes as well. Such a well-known (in the econometrics literature) example is an integrated GARCH process (D.B. Nelson, 1990, Stationarity and Persistence in the GARCH(1,1) Model, Econometric Theory 6: 318-34). Thus the caveat (strict stationarity does not imply weak stationarity) is certainly in order. The new sentence "Any strictly stationary process which has a mean and a covariance is also WSS." is fine. However, I think that the present descriptions of strict and weak stationarity (in Wikipedia) still inadequately suggest dominance of strict stationarity over weak stationarity (e.g. "A weaker form of stationarity commonly employed - - "). It is possible that a process is weakly stationary while not strictly stationary. This could happen if the first two moments were time invariant while the third or fourth, say, moment were not. This should be pointed out in the text as well IMO. (PJP) —The preceding unsigned comment was added by 128.214.205.4 (talk • contribs) 17 October 2006.

Thanks -- apologies for the revert, I wanted to get an example to be sure that what you were saying was correct. You are absolutely right that strict stationarity does not imply weak stationarity -- I had not realised this until now. Thanks for providing the example here. --Richard Clegg 15:12, 17 October 2006 (UTC)

Is it proper to equate wide-sense stationarity (WSS) with second-order stationarity. I think there is a difference. A process can be WSS without being second-order stationary. The definition in the article is that of WSS; second order stationary is different. (of course it depends on your definition, but that must be clarified.) See Section 6.2 of Peebles Jr., Peyton: Probability, Random Variables, and Random Signal Principles (2/e) Dakshayani 05:39, 4 November 2006 (UTC)

Could you quote that definition for us? I don't have the book. --Zvika 07:31, 4 November 2006 (UTC)

There is a difference between second order stationarity and WSS. A process is second order ( according to this def) if the second order density function satisfies for all . Such a process will be WSS if the mean and corelation functions are finite. A process can be WSS without being Second Order Stationary. The definition of Second Order Stationarity can be generalised to Nth order and strict stationary means stationary of all orders. Dakshayani 09:56, 4 November 2006 (UTC)

You're probably right then. Sounds like this is related to the previous discussion (above). I removed second-order stationarity from the definition. --Zvika 13:03, 4 November 2006 (UTC)