Talk:Stable distribution

Stable?
Nice article...

Are these the same as stable distributions? Clearly, they have the addition property, but are all "stable" distributions also "Levy skew alpha" stable d's? If so, material of stable distributions could be included - as an easy intro --J heisenberg 11:01, 21 Feb 2005 (UTC)

Hello - Yes, all stable distributions are included in what is being called here "Levy skew alpha" stable distributions. Some authors call them just stable distributions and call the case for alpha=3/2 the Levy distribution. Check out Nolan's web page. We need to get some plots of the skew distributions here too. Paul Reiser 14:22, 21 Feb 2005 (UTC)

I've included the stuff from the article above. Feel free to edit--J heisenberg 19:18, 21 Feb 2005 (UTC)

"yielding a distribution symmetric about c" <-- surely "...about mu"? -- Right. Its fixed, thanks. PAR 02:04, 4 August 2005 (UTC)

"Why if μ is the mode of the distribution, in the plot named 'Skewed centered Lévy distributions with unit scale factor' the x=0 is not the maximum?" dsalas


 * Good question - I fixed it. PAR 17:08, 24 September 2005 (UTC)


 * I've now redirected stable distributions to the singular, i.e. to this article. It is absurd to have that page redirecting to somewhere other than here. Michael Hardy (talk) 00:20, 21 February 2009 (UTC)

Merge

 * In probability theory, a stable distribution has the property of stability: If a number of independent identically distributed random variables have a stable distribution ...

Sigh, classic Wikipedia lead sentence ... let's start by getting rid of the "stability" article, shall we? --99.245.206.188 (talk) 02:16, 3 March 2009 (UTC)


 * agreed 160.83.73.25 (talk) 16:40, 1 September 2009 (UTC)


 * agree PAR (talk) 23:54, 15 February 2010 (UTC)


 * agree TomyDuby (talk) 02:32, 16 February 2010 (UTC)


 * I have removed the merge template for reasons stated at Talk:Stability (probability). If we have to discuss merging, then let's do it in one place and in the context of the present versions. Melcombe (talk) 12:49, 19 January 2011 (UTC)

sign?
Can someone please explain to me what sign(t) means in the expresion for $$\varphi (t)$$ Thanks in Advance.


 * Its just the sign of t, +1 for positive, -1 for negative, 0 for 0. I fixed it so it links to the sign function. PAR 20:19, 18 December 2005 (UTC)

Skewness
The skewing factor is indeed $$\beta tg({\pi \over 2}\alpha)$$, so the skewness is limited through $$\alpha$$, even if $$\beta$$ is always in the range [0,1]. I am not sure if this deserves an other plot ($$\beta$$=const>0, $$\alpha$$=1->2), but perhaps it should be noted explicitely. al

Cumulative function
Is there an expression for the cumulative function when beta=0? The sidebar says that it usually isn't analytically expressible, see the text; but it isn't discussed in the text. Bubba73 (talk), 22:21, 21 June 2006 (UTC)


 * I think that was referring to the special cases when the distribution becomes a normal distribution (&beta; has no effect) and the Cauchy distribution (&beta; =0). PAR 10:36, 12 November 2006 (UTC)

Some problems of the regression of Levy distribution
I've learnt about some skills of regressing the Levy tail distribution. They are powerlaw fit (double log fit), Hill estimator(and its variations), and empirical function approximation. I wonder if there is any else method, and which one is better, why? Is there any idea about this issue? Thanks.

Confusion
OK, I've taken probability classes at MIT no less, and I'm unable to figure out what a Levy distribution is after reading this article. I could probably do so after reading it a few more times, but this seems to be an indication that it is not well written for a broad technical audience. The introduction is particularly opaque. It uses α without giving a formula in which α is used, or any definition of what it might mean. "have the same distribution as the original." is a confusing phrase. What is "the original"? Is this saying that X1 and X2 are the same, and that the linear sum of the two is the same as both X1 and X2? The Y equations seem to indicate that the sum is merely of the same form. Or is that the same thing after normalizing to a zero-to-one scale? The graphics are also a little confusing, perhaps because the formula has not yet been given (reading the article top to bottom generally). At first I thought this was an example of multiple distributions being added, but I infer now it's just a bunch of sample functions that satisfy the distribution formula. It would help moving these lower so they appear after the forumla is given, but it would likely help even more showing sample X1 and X2 probability functions - one set that satisfies the additive constraint, and perhaps one that does not? -- Beland 02:52, 6 December 2006 (UTC)


 * Well, I took a try at fixing it. Is this an improvement? PAR 06:49, 6 December 2006 (UTC)

the range of alpha
in the text, it says [0,2], in the table it says (0,2). Is it very trivial, or some correction is needed? Jackzhp 02:04, 13 July 2007 (UTC)


 * I don't see where it says [0,2], but please change it if it does. It should be (0,2]. The distribution becomes a normal distribution when &alpha;=2 and in the limit of &alpha;=0 it becomes a Dirac delta function, which is undefined at t=0. PAR 03:50, 13 July 2007 (UTC)

Series representation of the pdf
As far as I can see the formula for the series representation only makes sense when $$\alpha\neq 1$$, otherwise, the $$\Phi$$ term contains the variable $$t$$, so the integration would not produce the formula stated. Am I missing something? —Preceding unsigned comment added by 192.193.245.16 (talk) 19:17, 17 January 2008 (UTC)
 * As far as I can see, you can't be missing something! The variable of integration — $$dt$$ — for a definite integral can't possibly be a variable — $$t$$ — in the result! Especially as it's singular at one end of the range of integration.
 * Adding to the confusion: it seems the $$dt$$ got dropped from the integral when the Taylor series was introduced:
 * $$f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[\int_0^\infty e^{it(x-\mu)}\sum_{n=0}^\infty\frac{(-qt^\alpha)^n}{n!}\right]dt\Leftarrow (missing\ dt)$$
 * While it's not all that unclear, we should be consistent in using proper notation.
 * Bob Kerns (talk) 12:27, 20 January 2008 (UTC)

In fact it is incorrect (even with $$\beta = 0$$, which avoids the problem described above) if $$\alpha > 1$$ as the series does not converge. If $$\alpha = 1$$ then it only converges for $$x$$ sufficiently large. $$|x-\mu| > |q|^{1/\alpha}$$, I think. Daren Cline (talk) 22:30, 19 March 2019 (UTC)

The solution given further down still has $$q$$, which is a function of $$t$$. So, the final series makes no sense. 104.187.53.82 (talk) 01:41, 11 January 2024 (UTC)

Improving the plots
Suggestion: It would be easier to grasp the paired plots, if the one curve in common between each pair, $$\alpha = 0.5,\mbox{ }\beta = 0.0$$, shared the same color. I'd suggest reversing the color sequence in the second plot, because the red tends to stand out best, (at least to those of us who are not color-blind). Making that one heavier would also help, and make it more accessible to those with red-green color-blindness.

Basically, the second plot takes the one curve from the first plot, and skews it. But to detect this, you have to search the curves for one of the same shape in both, and then verify your discovery by checking the key.

Ideally, the only color in common would be the curve in common. Perhaps differentiate the skew curves by decreasing saturation or brightness with increasing skew (i.e. fade to gray or black). Bob Kerns (talk) 11:46, 20 January 2008 (UTC)

Characteristic function error
Found and corrected an error in the characteristic function, the (|ct|2) term needed to be (|ct|2)/2; see history -> Nowaket. However, the symmetric probability density plots (upper right corner) were generated by the Fourier transform of the incorrect characteristic function. Clearly the apex of a zero mean, variance 1, Gaussian (α = 2) PDF should be ~0.4 not <0.3. Making this change corrects the PDF's.

Nowaket (talk) 13:57, 28 March 2008 (UTC)


 * No - please read the article before editing it. For c=1, the variance is $$\sigma^2=2c^2$$ and so the peak is at $$1/\sigma\sqrt{2\pi}$$=0.28... Also, please leave comments at the bottom of the talk page, not the top. PAR (talk) 18:12, 28 March 2008 (UTC)

Special Cases
Hi all, I found in the 'special cases' section that somebody (2006-08-15 10:18:02 IP: 163.156.240.17) states that: "...any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to unity)." As the IP address seems to belong to "AXA Technology Services UK Ltd.", I have no real way of contacting the author of this comment, but I have been able to prove the result myself (using a similar approach as Lee, Hopcraft and Jakeman here)

Does anybody know where the author of this got this from? I'm in the process of writing a paper which will contain this result using my proof, and it'd be nice to have the 'original' proof in there.

(edit) I've also got (and have verified using numerical methods such as Nolan's program) ten closed-form expressions for the densities of stable distributions, in terms of Whittaker functions, Fresnel integrals, modified Bessel functions, hypergeometric functions and Lommel functions. Is it okay to put these on the main page (perhaps in a closed-form expansions section or something), or should these be left off? I don't really want to make the page too complicated, but I think it'd be nice for researchers such as myself to have somewhere that they're written down explicitly. Thanks all! —Preceding unsigned comment added by Wainson (talk • contribs) 21:56, 4 August 2008 (UTC)


 * Wainson - yes please, do post those. Sounds very interesting.  Perhaps here in talk for a look-over, then move to main page?  Thanks!! ObsidianOrder (talk) 16:49, 14 August 2008 (UTC)

I post all distributions with scale factor 1, and location parameter 0, as it is easy to transform the expressions for arbitrary values of those parameters:

$$ f\biggl(x;\frac{1}{3},0,1,0\biggr) = Re\biggl(   \frac{2 \exp(- i \pi /4)}{3 \sqrt{3} \pi}    x^{-3/2}    S_{0,1/3} \Bigl(\frac{2 \exp(i \pi /4)}{3 \sqrt{3}} x^{-1/2} \Bigr) \biggr) $$

where $$S_{\mu,\nu}(z)$$ is a Lommel function

Source: T. M. Garoni, N. E. Frankel, "Lévy flights: Exact results and asymptotics beyond all orders", Journal of Mathematical Physics 43 #5, 2670-2689 (2002)

$$f\biggl(x;\frac{1}{2},0,1,0\biggr) = \frac{\vert x \vert^{-3/2}}\Biggl(\sin\left(\frac{1}{4\vert x \vert}\right)\Biggl[\frac{1}{2}-S\biggl(\sqrt{\frac{1}{2\pi\vert x \vert}}\biggr)\Biggr]+\cos\left(\frac{1}{4\vert x \vert}\right)\Biggl[\frac{1}{2}-C\biggl(\sqrt{\frac{1}{2\pi\vert x \vert}}\biggr)\Biggr]\Biggr)$$

where $$S(x)$$ and $$C(x)$$ are Fresnel Integrals.

Source: K. I. Hopcraft, E. Jakeman, R. M. J. Tanner, "Lévy random walks with fluctuating step number and multiscale behavior", Physical Review E 60 #5, 5327-5343 (1999)

$$f\biggl(x;\frac{2}{3},0,1,0\biggr) = \frac{1}{2\sqrt{3\pi}}\vert x \vert ^ {-1} \exp\biggl(\frac{2}{27}x^{-2}\biggr) W_{-1/2,1/6}\biggl(\frac{4}{27}x^{-2}\biggr)$$

where $$W_{k,\mu}(z)$$ is a Whittaker function.

Source: V. V. Uchaikin, V. M. Zolotarev, "Chance And Stability - Stable Distributions And Their Applications" - VSP, Utrecht, Netherlands (1999)

$$ f\biggl(x;\frac{4}{3},0,1,0\biggr) = \frac{3^{5/4}}{4 \sqrt{2 \pi}} \frac{\Gamma (7/12) \Gamma (11/12)}{\Gamma (6/12) \Gamma (8/12)} \,_2F_2 \biggl( \frac{7}{12}, \frac{11}{12}; \frac{6}{12}, \frac{8}{12}; \frac{3^3 x^4}{4^4} \biggr) - \frac{3^{11/4}x^3}{4^3 \sqrt{2 \pi}} \frac{\Gamma (13/12) \Gamma (17/12)}{\Gamma (18/12) \Gamma (15/12)} \,_2F_2 \biggl( \frac{13}{12}, \frac{17}{12}; \frac{18}{12}, \frac{15}{12}; \frac{3^3 x^4}{4^4} \biggr) $$
 * Something look awry here. Alpha=4/3, beta=0 should by symmetric, yet there is an x^3 term. I am also unable to numerically confirm the value at zero.  104.187.53.82 (talk) 23:49, 30 July 2023 (UTC)

Source: T. M. Garoni, N. E. Frankel, "Lévy flights: Exact results and asymptotics beyond all orders", Journal of Mathematical Physics 43 #5, 2670-2689 (2002)

$$ f\biggl(x;\frac{3}{2},0,1,0\biggr) = \frac{1}{\pi} \Gamma (5/3) \,_2F_3 \biggl( \frac{5}{12}, \frac{11}{12}; \frac{1}{3}, \frac{1}{2}, \frac{5}{6}; - \frac{2^2 x^6}{3^6} \biggr) - \frac{x^2}{3 \pi} \,_3F_4 \biggl( \frac{3}{4}, 1, \frac{5}{4}; \frac{2}{3}, \frac{5}{6}, \frac{7}{6}, \frac{4}{3}; - \frac{2^2 x^6}{3^6} \biggr) + \frac{7 x^4}{3^4 \pi ^ 2} \Gamma (4/3) \,_2F_3 \biggl( \frac{13}{12}, \frac{19}{12}; \frac{7}{6}, \frac{3}{2}, \frac{5}{3}; - \frac{2^2 x^6}{3^6} \biggr) $$

Source: T. M. Garoni, N. E. Frankel, "Lévy flights: Exact results and asymptotics beyond all orders", Journal of Mathematical Physics 43 #5, 2670-2689 (2002) --Wainson (talk) 14:35, 24 August 2008 (UTC)

The following are asymmetric distributions (specifically, where $$\beta = 1$$).

$$f\biggl(x;\frac{1}{3},1,1,0\biggr) = \frac{1}{\pi} \frac{2\sqrt{2}}{3^{7/4}}x^{-3/2}K_{1/3}\Biggl(\frac{4\sqrt{2}}{3^{9/4}}x^{-1/2}\Biggr)$$

where $$K_v(x)$$ is a modified Bessel function of the second kind.

Source: K. I. Hopcraft, E. Jakeman, R. M. J. Tanner, "Lévy random walks with fluctuating step number and multiscale behavior", Physical Review E 60 #5, 5327-5343 (1999)

$$f\biggl(x;\frac{2}{3},1,1,0\biggr) = \frac{\sqrt{3}}{\sqrt{\pi}}\vert x \vert ^ {-1} \exp\biggl(-\frac{16}{27}x^{-2}\biggr) W_{1/2,1/6}\biggl(\frac{32}{27}x^{-2}\biggr)$$

Source: V. M. Zolotarev, "Expression of the density of a stable distribution with exponent alpha greater than one by means of a frequency with exponent 1/alpha", In Selected Translations in Mathematical Statistics and Probability 1, 163-167 (1961). Translated from the Russian article: Dokl. Akad. Nauk SSSR. 98, 735-738 (1954)

$$f\biggl(x;\frac{3}{2},1,1,0\biggr) = \left\{ \begin{array}{ll} \frac{\sqrt{3}}{\sqrt{\pi}}\vert x \vert ^ {-1} \exp\biggl(\frac{1}{27}x^3\biggr) W_{1/2,1/6}\biggl(- \frac{2}{27}x^3\biggr) & x<0\\ \frac{1}{2\sqrt{3\pi}}\vert x \vert ^ {-1} \exp\biggl(\frac{1}{27}x^3\biggr) W_{-1/2,1/6}\biggl(\frac{2}{27}x^3\biggr) & x \geq 0 \end{array} \right.$$

Source: I. V. Zaliapin, Y. Y. Kagan, F. P. Schoenberg, "Approximating the Distribution of Pareto Sums", Pure and Applied Geophysics 162 #6, 1187-1228 (2005) --Wainson (talk) 00:10, 20 August 2008 (UTC)

The remaining three distributions are $$f\biggl(x;1,0,1,0\biggr)$$ (the Cauchy), $$f\biggl(x;\frac{1}{2},1,1,0\biggr)$$ (the Lévy) and $$f\biggl(x;2,0,1,0\biggr)$$ (the Gaussian).

Turns out I know of eleven. :) --Wainson (talk) 12:25, 20 August 2008 (UTC)

More symmetric distributions The symmetric distributions for which $$\alpha=p/q$$ and $$p>q$$ can be derived from a result in:

T. M. Garoni, N. E. Frankel, "Lévy flights: Exact results and asymptotics beyond all orders", Journal of Mathematical Physics 43 #5, 2670-2689 (2002)

--Wainson (talk) 12:16, 20 October 2008 (UTC)

Does anyone feel that these special cases are worthy of inclusion in the main article? PAR (talk) 21:06, 22 August 2011 (UTC)


 * I think so (and actually $$f\biggl(x;\frac{3}{2},0,1,0\biggr)$$ is mentioned in this article and has its own article). But I am concerned that adding lots of formulae with non-elementary functions might overwhelm the article with information of limited interest to most readers.  So it may be better to have a short section noting that there are other closed form density functions that use non-elementary functions with a link to a new article listing the circumstances and their densities and any other relevant information. Rlendog (talk) 21:27, 22 August 2011 (UTC)

Series representation does not converge
I ran into a problem with the formula given in "Series representation". It does not converge, pretty much ever? For an explanation of why, consider $$\Gamma(n+1) = n!$$, ergo if alpha=1 for example $$\frac{\Gamma(\alpha n+1)}{n!}$$ is always equal to 1, and if alpha > 1 then it grows with increasing n. On top of that if x - mu < 1, then as n gets larger, $$\left(\frac{i}{x-\mu}\right)^{\alpha n+1}$$ also gets arbitrarily large. I'm not sure what is causing it or what the problem with the derivation is. Anyone? ObsidianOrder (talk) 16:37, 14 August 2008 (UTC)


 * I'm not sure, but it might be an asymptotic expansion, which will never converge, but will come close (sometimes VERY close) to the functional value given. PAR (talk) 01:20, 20 August 2008 (UTC)


 * There is indeed a problem. If we follow "Chance and Stability, Stable Distributions and their Applications" by Uchaikin and Zolotarev (MR1745764), from pages 106, 108, and 109 we can recover different series expansions for when $$\alpha <1 $$, $$\alpha >1 $$ and $$\alpha =1 $$ respectively. This does solve the issue of non-convergent series.JorgeIGC (talk) 10:18, 10 March 2018 (UTC)


 * That's not the only problem. See "Series Representation" further up this talk page. If $$\beta \ne 0$$ then the expression does not even make sense as $$q$$ depends on the $$t$$ that supposedly was integrated out. Daren Cline (talk) 22:55, 19 March 2019 (UTC)


 * In fact, although the series converges if $$\alpha<1$$, Fubini's theorem still does not apply so "reversing the order of integration and summation" is inappropriate in any case. An accurate and responsible result is needed here, with citation. Daren Cline (talk) 12:38, 20 March 2019 (UTC)

Entropy
The infobox states 'see text' and there's nothing in the text about entropy. Solar Apex (talk) 01:35, 11 December 2008 (UTC)

Move to "Lévy Distribution"
It appears the article on "Lévy skew alpha-stable distribution" was moved by the user Ptrf to the Lévy distribution article on 20 Feb 2009, yet there is no discussion of the move. The move expunged a great deal of useful information which isn't available in the "Lévy Distribution" article. The history doesn't allow an undo. I am curious why it was moved, or at least why it was moved without incorporating the information present in the "alpha-stable" article into the "Lévy Distribution" one. Can anyone advise and suggest where to find the old "Lévy skew alpha-stable distribution" (other than through Google Cache) so that we may at the very least restore it. Imlepid (talk) 06:47, 21 February 2009 (UTC)


 * For the record: "Levy skew alpha-stable distribution" was moved to "stable distribution" (and not "Levy distribution"). I think there was a confusion in terminology that arose after a merge in 2005. Now, I believe the titles correspond to the content and, more importantly, to what is stated in (Nolan 2005) or NolanWeb1 pdf. You're welcome to check it. In general, moving pages results in no loss of information, it's possible, however, that some further adjustments might be needed (you're welcome to go for it). No page was lost as you seem to imply, just renamed. I hope this addresses your concerns. ptrf (talk) 10:57, 21 February 2009 (UTC)

Degenerate distribution
Can the degenerate distribution be considered an example of a stable distribution albeit a trivial one? Maybe I am missing something, but it seems to satisfy the condition. Also, its characteristic function seems almost, though not quite, identical to what a stable distribution characteristic function would be for alpha equal to zero, and would require only a minor tweak to the definition of the stable characteristic function (basically using a zero in the exponent when you have zero to zero power) to make it work. Rlendog (talk) 16:35, 28 July 2009 (UTC)

Range of c
c only enters as an absolute value in the characteristic function, so it makes no sense to limit c to the interval $$[0,\infty)$$. Either c should be so limited and the absolute value removed, or put c in the interval $$(-\infty,\infty)$$ and keep the absolute value. The latter is standard, and there are reasons for this (see for example McCullagh's parametrization of the Cauchy distributions).

(Allowing negative c adds no value and makes no physical sense. What is a negative scale factor? It might make sense if -c lead to change of sign of beta, but it doesn't. I vote for removal of the absolute-values. 104.187.53.82 (talk) 06:43, 25 July 2023 (UTC))

Relation to elliptical distributions
I'd like to see something about the relation of stable distributions to elliptical distributions. I know that a linear combination of jointly elliptical variables is elliptically distributed--this seems very similar to the definition of a stable variable. See

Chamberlain, G. 1983."A characterization of the distributions that imply mean-variance utility functions", Journal of Economic Theory 29, 185-201.

and

Owen, J., and Rabinovitch, R. 1983. "On the class of elliptical distributions and their applications to the theory of portfolio choice", Journal of Finance38, 745-752. Duoduoduo (talk) 20:15, 11 March 2010 (UTC)


 * The property that you're taking about is infinite divisibility, where any probability distribution can be formed as the sum of N distributions of the same class (where N is any strictly positive integer). Other such infinitely divisible distributions are the Poisson, Gaussian (Normal) and Gamma distributions, and of course, the stable distributions. Wainson (talk) 16:11, 16 May 2010 (UTC)


 * Wainson, that's incorrect. It is not about infinite divisibility. For example a linear combination of poisson variables isn't poisson, even though the poisson is infinitely divisible. The confusion here is that linear combination of *jointly* elliptical variables is elliptically distributed, while a linear combination of *independent* stable variables is stable distributed. The key thing to note here is that jointly elliptical variables are only independent in the case of the multivariate normal with diagonal covariance matrix. Finally, note that stable variables with a skewness parameter of 0 are also elliptical. Merudo (talk) 22:24, 20 May 2014 (UTC)

Limit of power tail distributions
The following statement seems incomplete or incorrect: "A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | α + 1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution f(x;α,0,c,0) as the number of variables grows. (Voit 2003, § 5.4.3)" For example, sum of random variables from a Cauchy distribution with a non-zero median will tend to a Cauchy distribution with a non-zero median, not to a zero median. Similarly, I am not sure a sum of random variables from an asymmetric distribution will necessarily tend to a symmetric stable distribution - a Levy distribution at least should not. Rlendog (talk) 17:34, 11 April 2011 (UTC)

(Continuous) stable distributions and (discrete) stable distributions
I was looking for information on the discrete-stable distributions when I was doing research for my thesis, but there was no page.

I'm in the process of writing one (see my draft), but don't know how best to link the articles together.

I'm thinking the best way might be to have a 'stable distributions' page which talks about general properties (e.g. limits of sums of iid random variables, being infinitely divisible, etc. then linking to the existing 'stable distributions' page (which will be renamed to 'continuous-stable distributions') and to my draft, 'discrete-stable distributions' from the general page.

How does that sound? Wainson (talk) 11:00, 20 April 2014 (UTC)


 * I think it's a great idea. PAR (talk) 10:59, 21 May 2014 (UTC)

Paul Levy
Mandelbrot claimed the name stable distribution comes from Paul Levy work:

''"...The purpose of this paper will be to present and test such a new model of price behavior in speculative markets. The principal feature of this model is that starting from the Bachelier process as applied to log,Z(t) instead of Z(t), I shall replace the Gaussian distributions throughout by another family of probability laws, to be referred to as "stable Paretian," which were first described in Paul Levy's classic Calcul des probabilites (1925). In a somewhat complex way, the Gaussian is a limiting case of this new family, so the new model is actually a generalization of that of Bachelier." ''

http://www.webofstories.com/play/benoit.mandelbrot/49

I can't read french, can someone confirm this?Lbertolotti (talk) 18:32, 1 October 2014 (UTC)

Levy used the term "stable lois" Lbertolotti (talk) 00:10, 13 October 2014 (UTC)

Copies versus data
This article uses the word "copies" in place of "data", as in realizations from a distribution. Is this a standard synonym in probability speak? Isambard Kingdom (talk) 01:42, 26 April 2015 (UTC)


 * A copy of a random variable seems to be another random variable with the same distribution (not data). That's the first time I see copy used in this way. Still, it may be "lingo". I have tried to improve the first sentence in the article and in doing so eliminated "copy". --Herbmuell (talk) 02:59, 21 June 2017 (UTC)

Merge proposal 2016
It has been a long time since the various merge and deletion proposals, and Stability (probability) still seems redundant, and relatively unsourced compared to Stable distribution. Is there any reason to keep the former? Dicklyon (talk) 06:01, 4 January 2016 (UTC)
 * Oppose - This article is about a family of probability distributions, the other article is about a property that this family exhibits. But since the former article also covers topics such as geometric stability, which this family of distributions does not exhibit (that would be the Geometric stable distribution), and max stability (and there are other forms of stability that could be covered in the stability article which are not appropriate to these stable distributions), I don't think a merge would be a good idea. Rlendog (talk) 15:16, 4 January 2016 (UTC)

Mellin transform and simulation
I think a section dedicated to stable distributions' Mellin transforms and the simulation algorithm derived from it is relevant. My main reference for what I'm about to write is "Simulation and Properties of Stable Law" by Devroye and James (MR3233961). Here is a quick overview of how it can be done: We explain that "stable distribution" is often known as "weakly stable distribution" whereas the "strongly stable distribution" has no shift or scaling (for which I prefer Zolotarev's (C) form parametrization since it makes everything much easier here and is a continuous parametrization of $$\varphi$$ (see Remark 1 and 2 in the cited article above). Here we may even make a distinction of when "linear shift" (when $$\alpha \neq 1$$) and a "logarithm shift" (when $$\alpha=1$$).

Now I will only talk about the case of strongly stable distributions (any $$\alpha \in (0,2]$$). The weakly stable cases follow by doing a linear transformation when $$\alpha\neq1$$ or with other procedures (also available in the article) when $$\alpha=1$$.

We then rewrite the characteristic function as:

$$\varphi(t;\alpha,\beta)=\exp\left( -|t|^\alpha \exp\left( -i\sgn(t)\frac{\alpha\theta\pi}{2}\right)\right)$$

where $$\theta = \beta \left(\frac{\alpha-2}{\alpha}1_{\alpha>1}+1_{\alpha\leq1}\right)$$. Next, if we put $$\rho=\frac{1+\theta}{2}$$ and let $$X$$ be a stable random variable with that characteristic function, then for any $$s\in\mathbb{C}$$ with $$-1<\Re s<\alpha $$:

$$E\left[X^s1_{X>0}\right]= \rho\frac{\Gamma(1+s)\Gamma(1-s/\alpha)}{\Gamma(1+s\rho)\Gamma(1-s\rho)}$$

In particular $$P(X>0)=\rho$$ and if $$\alpha>1$$ and $$\beta=-1$$ ("spectrally negative" case) this yields the previous formula valid for any $$s\in\mathbb{C}$$ with $$-1<\Re s $$. In that case, the distribution even has a moment generating function (wich can be expressed as a power series using this formula and we might add it for completeness).

Using the last relationship (the Mellin transform), then we deduce that $$X|X>0 $$ has the same distribution as $$\left(\frac{Y_1}{Y_2}\right)^{\rho} $$ where $$Y_1 $$ and $$Y_2 $$ are independent stable random variables with parameters $$(\alpha\rho,1) $$ and $$(\rho,1) $$ respectively. These random variables are easily simulated using the formula given in Devroye and James' article page 2.JorgeIGC (talk) 11:18, 10 March 2018 (UTC)

Original research in the generalization section
Everything after the first two paragraphs seems to be either original research or poorly cited. Given the content, I think it's the former, and a WP:NOR case; regardless, this section needs to be better set-out. I would delete the entire section, but I don't want my first major edit to be removing a large chunk of a page to fix an issue that could be solved with one or two citations. 124.171.219.139 (talk) 03:52, 30 April 2020 (UTC)

The majority of the GCLT section should be tossed. Why is there some random proof of another theorem here??? GLCT was proven (published) in 1937 by Levy and the statement and proof there is what is relevant. 137.79.234.22 (talk) 18:57, 24 February 2023 (UTC)


 * I added a new GCLT section with full references and a statement. I didn't blast the old section; I will leave that up to someone bolder. I did replace the nonsense over on the CLT page, though. Skewray (talk) 00:05, 26 June 2023 (UTC)


 * Section removed Skewray (talk) 16:04, 19 May 2024 (UTC)

Singular or plural?
The first sentence of the article is as follows:

"The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it."

Question: Is it correct to refer to the family of stable distributions as "the Lévy-stable distribution" (singular) ???

This would be most unusual.

Perhaps it should be the "Lévy-stable distribution family" ???

I hope someone knowledgeable about this subject can fix this apparent problem. 2601:200:C000:1A0:4C93:30C6:6EBC:CB9C (talk) 20:56, 23 August 2022 (UTC)

A Generalized Central Limit Theorem Isn't
The section on A Generalized Central Limit Theorem is only on an early and related proof. The GCLT says that all distributions when summed converge (in some sense) to Stable Distributions, and this isn't it. See Nolan's book.

104.187.53.82 (talk) 15:45, 4 November 2022 (UTC)


 * I added a new The GCLT section with full references and a statement. I didn't blast the old section; I will leave that up to someone bolder. I did replace the nonsense over on the CLT page, though. Skewray (talk) 00:06, 26 June 2023 (UTC)


 * Section removed Skewray (talk) 16:04, 19 May 2024 (UTC)