# Talk:Stability (probability)

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Field: Probability and statistics

All that is wrong (sorry), since an important condition is forgotten: the sum of INDEPENDENT random variables is meant.Boris Tsirelson (talk) 08:58, 28 September 2008 (UTC)

## Merge proposal (Feb 2009)

• Keep But reduce and extend. Let the stable distribution article be about the stable distribution as a distribution. Let this or a renamed article be about "stable laws". The article can then be reasonably extended to include the generalisations of the basic stable law included here to include the others included by Dodge in "The Oxford Dictionary of Statistical Terms" and which are presumably used elsewhere ... "symmetric stable law", "semi-stable law", "generalized stable law". Melcombe (talk) 14:27, 20 February 2009 (UTC)
• Week merge, as proposed, but it really depends on (answers to) the following questions. What extension of the main article stable distribution can be envisaged here? I'm afraid that "symmetric stable laws" are a (very) special case of stable laws, so it goes there. Generalisations have also its natural place in the main article. Further, I use law as a synonym of distribution (so does Nolan in his book, as it seems, and many others). I'm not sure what is meant by semi-stable law. ptrf (talk) 15:13, 20 February 2009 (UTC)
A semi-stable law is said, in the Dictionary mentioned, to be one for which the distribution function satisfies
φ(t)= φ(βt)γ ,
for all t, with γ > 0 and 0 < |β| <1. Unfortunately no reference is given. I think the point about "law" and "distribution" is that one thinks up some law that a distribution should obey and then finds the distributions that do obey that law. Melcombe (talk) 16:42, 20 February 2009 (UTC)
OK, let's put the distinction law-distribution aside. As for the merge proposal, if the Dictionary lists sufficiently many laws/entries not making individual Wikipedia articles, one may think about something like stable and related laws article. If not, the merge would be in order and possibly a couple of dictionary entries would find its place in the main article. I leave it to you. ptrf (talk) 17:01, 20 February 2009 (UTC)
• Merge I also use law as a synonym of distribution, and moreover I suspect that Dodge in "The Oxford Dictionary of Statistical Terms" does the same (according to the quotes given above by Melcombe). About "law that a distribution should obey", I think, one says rather "property that a distribution (or law) has", or "assumption that a distribution (or law) satisfies". If this article contains something really missing in the "stable distribution" article, it can be easily moved thereto. Boris Tsirelson (talk) 18:42, 20 February 2009 (UTC)

And, by the way, independence is still forgotten in the section "Calculating the PDF for the linear combination". Boris Tsirelson (talk) 19:06, 20 February 2009 (UTC)

Right; while rewriting some parts I discovered the other article; the further development may depend on the decision on the merger... ptrf (talk) 20:57, 20 February 2009 (UTC)
On reflection, there are more problems. For example, the article uses PDFs that in general need not exist until the distribution is shown to be stable... So I restored the tag. ptrf (talk) 21:18, 20 February 2009 (UTC)

## Supression proposal

Here goes a short review of the article.

• Intro is not too bad but more or less contained in the "main" article (stable distribution).
• the Definition section is lengthy and uses somewhat cumbersome language and problematic approach (e.g. existence of the densities prior to definition)
• calculating PDF of a convolution is not particularly specific to this topic (so of no interest). BTW, the section enjoys some innacuracies
• Examples are not representative (gaussian only, no big interest)
• The relationship for mu and c are (nik greater generality) contained in the "main" article. The same for sources/references.

It seems that this article may be safely deleted and no merge is really needed. If there are some deas as for "stable-related" topics, they'd be better written from scratch than based on this. Comments? ptrf (talk) 17:06, 21 February 2009 (UTC)

I agree. Boris Tsirelson (talk) 07:06, 22 February 2009 (UTC)
I agree that most of this should be deleted. The second paragraph at least says something specific about the importance of the topic might be a start for a new intro to an improved version of what is in stable distribution. However there is still far too much duplication between Lévy distribution and stable distribution. I would prefer to have the first made in something about the distribtion, to fit into Category:Continuous distributions while the second would have content more suited to Category:Theory of probability distributions. A start could be made by renaming stable distribution to "Stability (probability)", so as to replace this one, and by moving and or copying some of the diagrams into the Lévy distribution where they would help to form a more complete set. Melcombe (talk) 11:44, 23 February 2009 (UTC)
(Brief hold please)I have placed a pointer to this discussion in Wikipedia talk:WikiProject Statistics to invite more discussants. But I think only waiting 1 or 2 days is warranted. Melcombe (talk) 11:54, 23 February 2009 (UTC)

I have removed the prod tag, which was added even though there was obviously not a consensus, and replaced what was here by a very cut down version which should meet concerns over accuracy and duplication. However there does need to be a distinction between the stability property and the stable distributions. There also needs to be scope for indicating/discussing the connection with related ideas such as those in the "see also"s. Melcombe (talk) 16:18, 24 February 2009 (UTC)

Ooops, sorry for the prod; somehow I managed to miss your hold-on posts. Certainly, I have nothing against more "stability" on the Wikipedia. Still, there is a question whether the content justifies two separate articles or we can pack everything into "stable distribution". For the moment I prefer the latter, but this may change. Others may have different ideas. ptrf (talk) 16:58, 24 February 2009 (UTC)
The new text is good. About "two separate articles": I tried seeking precedents, and here is what I see. Continuity (mathematics) redirects to Continuous function. On the other hand, linearity exists separately from linear function. However, (a) linearity is treated much wider than just "the property of a function of being linear", and (b) linearity is suggested to be split. Boris Tsirelson (talk) 19:25, 24 February 2009 (UTC)
Actually, we have two articles Continuous function and Continuous function (topology). The content obviously warrants separation. IMHO this is also clear for linearity (split or not) and linear function. I'm not 100% sure whether there is more to stability_(probability theory) than stable random distributions. But I'm open to discussion. ptrf (talk) 22:43, 24 February 2009 (UTC)

## Stability in probability theory - unclear to me

The new section "Stability in probability theory" is not clear to me. Probably it is because I did not read the book by Lukacs (or something equivalent). But that book should not be a prerequisite for reading this article.

In the lead it is said that a random variable has the "stability" property if and only if its distribution is "stable" (as defined in the "stable distribution" article). Right? In other words, one class of distributions is the object of both articles. Right? Then, how to interpret the phrase "call these stable distributions, without meaning specifically the distribution described as the stable distribution" of the new section? It seems, the meaning of "stability" in the new section is different from that in the lead. Boris Tsirelson (talk) 14:14, 26 February 2009 (UTC)

In a mathematical context, one starts with some propositions describing what one is considering, and draws some conclusions by logical/mathematical arguments. The propositions and the conclusions are different things. Some of the (preliminary) conclusions that can be drawn from the propositions are stated briefly in this article. The meaning of stability is taken to mean what is said in this article, but it probably needs to be made clearer. Just because the article that is now named stable distribution was renamed from its original title does not mean it has to be a primary source.
I think there is still more to be said about the proposition of stability set in other contexts, although I don't know what results there are. The next most obvious thing is to think of multivariate distributions having a stability property, since the concept of convolution extends. But what about circular and spherical distributions: does the equivalent of convolution in these contexts yield any distributions having a stability property? Possibly only uniform distributions on these sets?
Melcombe (talk) 17:29, 26 February 2009 (UTC)
That could be an interesting research proposal (if not known already), but here in Wikipedia we do not research! The article should not be left in a self-contradictory state for a long time. Ideas where to look for more, and what kind of "more" could be found, are welcome on this talk page. However, as long as these ideas are not implemented, they should not enter the article. Right?
You claim (citing the book) that SUCH distributions are unimodal etc. For WHICH class of distributions is it stated in the book? On the sphere?!
Thus, I return to the question: in the article, as it exists today, what exactly is meant by stability?
I suggest you look at more existing articles, for example in list of statistics topics, and you we see that the present state of this one compares well. It is reasonable to make provision for extensions when you know that they can be made. Melcombe (talk) 11:31, 27 February 2009 (UTC)

Just a sidenote: the article that is now named stable distribution was renamed from its previous title just because it seems the correct choice (see virtually any textbook treating stable distributions). BTW, before 2005 there were two articles: "stable distribution" and "Levy skew alpha-stable". They were merged and, unfortunately, the latter title was retained instead of the former. No "primary source" was thus created by the recent move, it was just an attempt to arrange the things as they go in standard sources.

If you find sources treating "stability" in a wider sense or in a more abstract way than "stable distributions" (not that bad article, BTW), feel free to develop stability (probability). ptrf (talk) 10:30, 27 February 2009 (UTC)

Well the name change rather broke the connection with the Levy distribution article which conatins the following "connection"
• Relation to stable distribution: If $X \sim \textrm{Levy}(c)$ then $X \sim \textrm{Levy-S}\alpha\textrm{S}(1/2,1,c,0)$
but I am not sure that what is now stable distribution ever contained this notation. A similar thing is in Cauchy distribution. More thought needs to put into whether "stable distribution" is what is wanted in terms of its context of all the other "distribution" articles in subcategories Category:Continuous distributions and Category:Discrete distributions, which are moving towards a similar type of structure where there the functional form of the distribution is placed first and the context in which it arises it placed later, and connections to other distributions towards the end. This business of abbreviated names for distributions occuring in these "connections" seems partly specific to Wikipedia but many are much more widely used... are there common conventions for stable distributions? My guess is that previous mergings and renamings have been done thinking only about the single topic and not about how the contents need to relate to other articles.
For the present article there needs to be more about why stability is studied in probability theory and its context in the chain of ideas — decomposibility;infininte divisibility;stability. I see that Feller's Volume 2 has some stuff on multi-dimensional stable distributions, and it may also be a good source for context. I have also come across mention of "stable processes" in the conext of stochastic processes. Melcombe (talk) 11:31, 27 February 2009 (UTC)
At the very least, if unimodality (and some other properties) is stated for every distribution of some class, then it must be formulated clearly, which class of distributions is meant. And it must be clear, whether this class of distributions is equal to (or wider than, or narrower than, or whatever) the class of distributions treated in the "stable distribution" article. Or do I want too much? Or is it against the tradition of articles in statistics? In the last case probably we'll need to separate more clearly articles in statistics and articles in mathematics (including probability theory). Boris Tsirelson (talk) 11:56, 27 February 2009 (UTC)
I have put in something specific, linking to univariate distribution ... you won't like that article. Melcombe (talk) 17:24, 27 February 2009 (UTC)
I guess I second this. If there is no difference in the scope we may want to have only one article. If there exists a different approach/object/whatever, we may want to separate it. But should the latter be chosen, it needs an explicit justification (sources). At the moment it seems that the content may go to one article because no "stability" beyond "stable distribution" properties is presented. ptrf (talk) 14:48, 27 February 2009 (UTC)
So then, you'd move everything in central limit theorem into normal distribution ... the relationship is much the same. Melcombe (talk) 17:24, 27 February 2009 (UTC)
BTW, "a chain of ideas" has already been started as "generalized limit theorem" section on stable distributions page. Certainly, the Feller book (you mentioned) is much better source for this than (Voit 2003) cited in that article. This can and should be expanded (the section, not Feller;)). But, IMHO, it'd be confusing if we moved that section to a separate article. My present approach is, tentatively, let's add the content and we shall see what happens. We lack e.g. the Levy measure for stable distributions (and the Levy-Khinchin formula, see Feller), a major gap. Note that it goes naturally to the "main" article. ptrf (talk) 15:02, 27 February 2009 (UTC)

' Levy distribution article (..) conatins the following "connection"' - this is a good catch! Lévy distribution should be adjusted accordingly at some places. I think I agree that the previous merging was "ad-hoc" and some further things might need adjusting as well (still believe that what we have now is closer to cited sources). BTW, the notation(s) you mentioned is far from universal and thus should be explained or fixed (e.g SαS often means 'symmetric-alpha-stable', not 'skew-alpha-stable', which, in turn, is close to 'Levy completely skew alpha-stable', that is Lévy distribution itself... There was a real confusion in the names. Anyway, I'll try to fix it. It's good that "stable" articles get some attention). ptrf (talk) 14:48, 27 February 2009 (UTC)

I suppose the Levy–SαS notation might reasonably be replaced by "Stable" for these Wikipedia articles, unless there is anything more widely used. I don't know if it is possible to do a general search to find all articles in which "Levy–SαS" is contained as maybe stuff inside the math tags isn't searched. Melcombe (talk) 16:32, 27 February 2009 (UTC)

## New merge (Mar 2009)

Reviving this ... not really following what warrants the separation. From what I gather, some think that there is a potential to cover some specialisations or generalisations, but I don't see why they can't be mentioned in the main article. If the main article started to get really crowded, then we could reconsider at that time. --99.245.206.188 (talk) 02:34, 3 March 2009 (UTC)

I have removed the merge template of March 2009, as no apparent response and the existence of previous discussions relating the radically different articles makes things confusing. If anyone wants to restart discussion of this can we have a proper pointer od a new discussion section. Melcombe (talk) 12:34, 19 January 2011 (UTC)