# Talk:Stable distribution

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## 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

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)

• agree PAR (talk) 23:54, 15 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)

## References

This is a nice article to a topic where there's not too much readable information on the web. It would benefit from additional references, stating clearly which book or article the individual results are taken from. 134.155.68.246 11:04, 23 December 2005 (UTC)

I agree - I added explicit references that I am familiar with. PAR 16:09, 23 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 (β has no effect) and the Cauchy distribution (β =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 α=2 and in the limit of α=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)

## 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 (talkcontribs) 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)

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

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)

$f\biggl(x;\frac{4}{3},0,1,0\biggr) = \frac{3^{5/4}}{4 \sqrt{2 \pi}} \frac{\Gamma (7/12) \Gamma (11/12)}{\Gamma (6/12) \Gamma (8/12)} \,_2F_2 \biggl( \frac{7}{12}, \frac{11}{12}; \frac{6}{12}, \frac{8}{12}; \frac{3^3 x^4}{4^4} \biggr) - \frac{3^{11/4}x^3}{4^3 \sqrt{2 \pi}} \frac{\Gamma (13/12) \Gamma (17/12)}{\Gamma (18/12) \Gamma (15/12)} \,_2F_2 \biggl( \frac{13}{12}, \frac{17}{12}; \frac{18}{12}, \frac{15}{12}; \frac{3^3 x^4}{4^4} \biggr)$

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 fomulae 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)

## 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).

## 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)

## 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)