Talk:Stationary process

message: 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)
replies: text: I believe the , which has been up for two years, is outdated.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)
thread_title: Style

message: as of 25 july 2018, the intro is way way to technical for a general encylopedia
remember, your mom, or dad, has to be able to understand it
Michael Hardy is 110% correct  — Preceding unsigned comment added by 50.245.17.105 (talk) 20:43, 25 July 2018 (UTC)
thread_title: style

message: (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)
replies: text: 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)
replies: text: 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)
thread_title:  Stationary process in statistics 

message: The introduction strongly suggests that all stationary discrete-time random processes are Bernoulli. I don't think that is correct. For example, isn't a random process where the distribution of Xn is determined by Xn-k, Xn-k+1, ..., Xn-1 (in other words, an order k Markov chain) stationary?  —Preceding unsigned comment added by 77.162.102.4 (talk) 14:02, 21 December 2007 (UTC)
replies: text: This certainly looks like a mistake to me. I've gone ahead and fixed it, and also the related mistake in the page Bernoulli scheme. If anyone thinks differently, please correct me. --Zvika (talk) 19:32, 22 December 2007 (UTC)
replies: text: No, the original statement was correct. Just take the n'th product of the of the Markov chain, where n is the length of the longest period. I agree, though, additional detail would be useful. In short, though, the Bernoulli shift is isomorphic to pretty much everything, including many/most dynamical systems. See, for example  for one statement of this. Err, well, when its infinite, you just get the Koopman operator and its adjoint, the transfer operator; in either case, you're still dealing with a shift operator, of which Markov chains and subshifts of finite type are more-or-less, in a hand-waving way, are special cases. linas (talk) 17:21, 21 November 2010 (UTC)
thread_title:  not all stationary discrete-time processes on {0,1} are bernoulli processes?! 

message: A recent revision (no real objection to this) has diverted an attempt to make the intro material more accurate ... the present version seems to me to imply that only the marginal distribution need be fixed. Also, the rearrangement into sections means that weak stationarity is not mentioned till much later ... it would be good to have both weak and strict refered to in the intro. Can someone find some appropriate wording to deal with these points. Melcombe (talk) 11:11, 9 April 2008 (UTC)
replies: text: I realize now what you were trying to say. Sorry for reverting you, but I think the sentence you added was more confusing than helpful. The lead should be accurate, but not verbose. I tweaked it about just now -- tried to emphasize the fact that we're talking about the joint distribution. What do you think? --Zvika (talk) 11:52, 9 April 2008 (UTC)

text: I think the intro is still pretty poor.  "If you draw a line through the middle of a stationary process ..."   What does "through the middle" mean?  Ma-Ma-Max Headroom (talk) 19:19, 2 December 2022 (UTC)
thread_title:  more precision in intro? 

message: "However, the sound of a cymbal crashing [ like one hand clapping?! ] is not stationary because the acoustic power of the crash (and hence its variance) diminishes with time."  I don't think this is correct: The sound is not the process; the process is one that always reverts to zero decibels.  Seems like a stationary AR(1) process to me, with the occasional crashing being the additive stochastic term.  Duoduoduo (talk) 18:51, 2 December 2010 (UTC)
thread_title: The sound of a cymbal crashing

message: "Stationarity, is defined as a quality of a process in which the statistical parameters (mean and standard deviation) of the process do not change with time. The most important property of a stationary process is that the auto-correlation function (ACF) depends on lag alone and does not change with the time at which the function was calculated."
A process with mean 0 and standard deviation 1 may have very different ACFs. Thus, such a process may switch at some (nonrandom) time from one ACF to another ACF. Then it does satisfy the first phrase above, but violate the second one. Not a valid definition. I revert. Boris Tsirelson (talk) 17:09, 27 December 2012 (UTC)
thread_title:  Inconsistent "definition" 

message: The lede currently says
:Note that a "stationary process" is not the same thing as a "process with a stationary distribution".
But the link stationary distribution says 
: Stationary distribution may refer to...[t]he set of joint probability distributions of a stationary process....
So it seems to me that the passage in this article contradicts its own link. I'll remove it unless there is an objection. Duoduoduo (talk) 14:08, 11 January 2013 (UTC)
replies: text: Or maybe the article "Stationary distribution" should be changed? I never saw that notion applied to non-stationary processes, but someone could say that a nonlinear deterministic time change of a stationary process leads to a non-stationary process with a stationary (marginal, single-time) distribution... I am not sure. Boris Tsirelson (talk) 14:42, 11 January 2013 (UTC)
replies: text: This issue is still not resolved. 178.38.78.134 (talk) 00:38, 22 January 2015 (UTC)
thread_title:  "Stationary process" and "process with a stationary distribution" 

message: 1. This new paragraph treats the discrete-time case, while the rest of the article is about continuous time.
2. If the discrete time runs from -∞ to +∞ then "the Hilbert subspace of L2(μ) generated by {e-2πiλ⋅t}" is the whole L2(μ), isn't it?
Boris Tsirelson (talk) 11:18, 13 January 2013 (UTC)
replies: text: Fixed to be consistent with rest of article. My bad. Mct mht (talk) 23:28, 13 January 2013 (UTC)

text: Continuity of the (positive definite) autocovariation function can be violated, but only if the Hilbert space is nonseparable. Boris Tsirelson (talk) 16:55, 17 September 2014 (UTC)
thread_title:  Wide sense stationarity and Hilbert space techniques 

message: Is it just me, or are the examples involving Y misleading? I understand the context in which they are stationary, but I think they might do more to confuse than inform. For evidence, just look at the debate on this page. I'd suggest either removing them entirely, or adding some sectioning above them to give a clearer indication that they are (in my opinion) "pathological examples".
briardew (talk) 16:35, 20 November 2013 (UTC)
thread_title:  Examples 

message: as an example:
for me running Opera, the maths in:
     Formally, let \left\{X_t\right\} be a stochastic process...
looks all fucked up, wheras the maths in:
     and let F_{X}(x_{t_1 + \tau}, \ldots, x_{t_k + \tau}) represent the cumulative distribution function of the joint distribution of...
looks fine.
by all fucked up I mean lots of visual noise making the image almost unreadable.
woo-hoo my first constructive wikipedia edit, up until now i've just vandalised things ocasionally. — Preceding unsigned comment added by 2001:4c28:4000:721:185:26:182:33 (talk) 19:38, 10 October 2014 (UTC)
thread_title:  embedded maths notation formatting (possible Opera bug) 

message: To make the claim in the definition that \tau does not affect F_{X} one needs extra hypotheses on F_{X}.  — Preceding unsigned comment added by MMmpds (talk • contribs) 19:30, 22 August 2016 (UTC)
thread_title:  Hypotheses on the CDF Function 

message: I am puzzled with Section "Relation between types of stationarity". There I read: "If a stochastic process is wide-sense stationary and has finite second moments, it is wide-sense stationary." Does it make sense? Just A\land B \Longrightarrow A?
Also, the definition in Section "Weak or wide-sense stationarity" says that weak-sense stationarity, wide-sense stationarity (WSS), covariance stationarity, and second-order stationarity are just different names of a single notion. But Section "N-th Order Stationarity" does not require any moments. Does it mean that "2-th order stationarity" is not the same as "second-order stationarity"? Boris Tsirelson (talk) 20:29, 18 December 2018 (UTC)
replies: text: About the first issue. This is an error. I fixed it. Thank you for reporting. About the second issue, the problem here is that the terminology depends on the author as mentioned in the section Stationary process#Other terminology. I propose using the term wide-sense stationary in the whole article except in the section about other terminology. Fvultier (talk) 22:33, 18 December 2018 (UTC)
thread_title:  Puzzled 

message: Eq.2 and Eq.1 are exactly the same  — Preceding unsigned comment added by Nanoukaplus (talk • contribs) 13:15, 5 March 2019 (UTC)
thread_title:  N-th stationarity 

message: The three examples all have the same character of being dependent on a single random variable. This is a very specialized case. It would be nice to have some more realistic cases. 2A02:1210:2642:4A00:85A5:3302:64C8:5866 (talk) 18:33, 28 October 2023 (UTC)
thread_title:  Examples