Editing Stable distribution (section)

==Definition==
A non-[[degenerate distribution]] is a stable distribution if it satisfies the following property:

{{block indent | em = 1.5 | text = Let {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} be independent realizations of a [[random variable]] {{math|''X''}}. Then {{math|''X''}} is said to be '''stable''' if for any constants {{math|''a'' > 0}} and {{math|''b'' > 0}} the random variable {{math|''aX''<sub>1</sub> + ''bX''<sub>2</sub>}} has the same distribution as {{math|''cX'' + ''d''}} for some constants {{math|''c'' > 0}} and {{math|''d''}}. The distribution is said to be ''strictly stable'' if this holds with {{math|1=''d'' = 0}}.<ref name=":0">{{Cite web|url = http://academic2.american.edu/~jpnolan/stable/chap1.pdf|title = Stable Distributions – Models for Heavy Tailed Data|access-date = 2009-02-21|last = Nolan|first = John P.|archive-date = 2011-07-17|archive-url = https://web.archive.org/web/20110717003439/http://academic2.american.edu/~jpnolan/stable/chap1.pdf|url-status = dead}}</ref>}}

Since the [[normal distribution]], the [[Cauchy distribution]], and the [[Lévy distribution]] all have the above property, it follows that they are special cases of stable distributions.

Such distributions form a four-parameter family of continuous [[probability distribution]]s parametrized by location and scale parameters ''μ'' and ''c'', respectively, and two shape parameters <math>\beta</math> and <math>\alpha</math>, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures).

The [[Characteristic function (probability theory)|characteristic function]] <math>\varphi(t) </math> of any probability distribution is the [[Fourier transform]] of its probability density function <math>f(x) </math>. The density function is therefore the inverse Fourier transform of the characteristic function:<ref>{{Cite web|url=https://www.randomservices.org/random/special/Stable.html| title=Stable Distributions|last=Siegrist|first=Kyle|website=www.randomservices.org|language=en|access-date=2018-10-18}}</ref>
<math display="block"> \varphi(t) = \int_{- \infty}^\infty f(x)e^{ ixt}\,dx. </math>

Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be expressed analytically. A random variable ''X'' is called stable if its characteristic function can be written as<ref name=":0" /><ref name=":1">{{Cite book|title = The Statistical Mechanics of Financial Markets – Springer | doi = 10.1007/b137351|last = Voit|first = Johannes|editor4-first = W|editor4-last = Thirring|editor3-first = H|editor3-last = Grosse | editor2-first = W|editor2-last = Beiglböck|editor1-first = R|editor1-last = Balian|publisher = Springer|year = 2005 | series = Texts and Monographs in Physics|isbn = 978-3-540-26285-5}}</ref>
<math display="block"> \varphi(t; \alpha, \beta, c, \mu) =  \exp \left ( i t \mu - |c t|^\alpha \left ( 1 - i \beta \sgn(t) \Phi \right ) \right ) </math>
where {{math|sgn(''t'')}} is just the [[sign function|sign]] of {{mvar|t}} and
<math display="block"> \Phi = \begin{cases}
\tan \left (\frac{\pi \alpha}{2} \right) & \alpha \neq 1 \\
- \frac{2}{\pi}\log|t| & \alpha = 1
\end{cases} </math>
''μ'' ∈ '''R'''  is a shift parameter, <math>\beta \in [-1,1]</math>, called the ''skewness parameter'', is a measure of asymmetry. Notice that in this context the usual [[skewness]] is not well defined, as for <math>\alpha < 2</math> the distribution does not admit 2nd or higher [[moment (mathematics)|moments]], and the usual skewness definition is the 3rd [[central moment]].

The reason this gives a stable distribution is that the characteristic function for the sum of two independent random variables equals the product of the two corresponding characteristic functions. Adding two random variables from a stable distribution gives something with the same values of <math>\alpha</math> and <math>\beta</math>, but possibly different values of ''μ'' and ''c''.

Not every function is the characteristic function of a legitimate probability distribution (that is, one whose [[cumulative distribution function]] is real and goes from 0 to 1 without decreasing), but the characteristic functions given above will be legitimate so long as the parameters are in their ranges. The value of the characteristic function at some value ''t'' is the complex conjugate of its value at −''t'' as it should be so that the probability distribution function will be real.

In the simplest case <math>\beta = 0</math>, the characteristic function is just a [[stretched exponential function]]; the  distribution is symmetric about ''μ'' and is referred to as a (Lévy) '''symmetric alpha-stable distribution''', often abbreviated ''SαS''.

When <math>\alpha < 1</math> and <math>\beta = 1</math>, the distribution is supported on [''μ'', ∞).

The parameter ''c'' > 0 is a scale factor which is a measure of the width of the distribution while <math>\alpha</math> is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution.

=== Parametrizations ===
The parametrization of stable distributions is not unique. Nolan <ref name="Nolan2020" /> tabulates 11 parametrizations seen in the literature and gives conversion formulas. The two most commonly used parametrizations are the one above (Nolan's "1") and the one immediately below (Nolan's "0").

The parametrization above is easiest to use for theoretical work, but its probability density is not continuous in the parameters at <math>\alpha =1</math>.<ref name="Nolan 1997"/> A continuous parametrization, better for numerical work,  is<ref name=":0" />
<math display="block"> \varphi(t; \alpha, \beta, \gamma, \delta) = \exp \left (i t \delta - |\gamma t|^\alpha \left (1 - i \beta \sgn(t) \Phi \right ) \right ) </math>
where:
<math display="block"> \Phi = \begin{cases} \left ( |\gamma t|^{1 - \alpha} - 1 \right ) \tan \left (\tfrac{\pi \alpha}{2} \right )  & \alpha \neq 1 \\ - \frac{2}{\pi} \log|\gamma t| & \alpha = 1 \end{cases} </math>

The ranges of <math>\alpha</math> and <math>\beta</math> are the same as before, ''γ'' (like ''c'') should be positive, and ''δ'' (like ''μ'') should be real.

In either parametrization one can make a linear transformation of the random variable to get a random variable whose density is <math> f(y; \alpha, \beta, 1, 0) </math>. In the first parametrization, this is done by defining the new variable:
<math display="block"> y = \begin{cases} \frac{x - \mu}\gamma & \alpha \neq 1 \\  \frac{x - \mu}\gamma - \beta\frac 2\pi\ln\gamma  & \alpha = 1 \end{cases} </math>

For the second parametrization, simply use
<math display="block"> y = \frac{x-\delta}\gamma </math>
independent of <math>\alpha</math>. In the first parametrization, if the mean exists (that is, <math>\alpha > 1</math>) then it is equal to ''μ'', whereas in the second parametrization when the mean exists it is equal to <math> \delta - \beta \gamma \tan \left (\tfrac{\pi\alpha}{2} \right).</math>

===The distribution===
A stable distribution is therefore specified by the above four parameters. It can be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function.<ref name=":0" /> If <math> f(x; \alpha, \beta, c, \mu) </math> denotes the density of ''X'' and ''Y'' is the sum of independent copies of ''X'':
<math display="block"> Y = \sum_{i = 1}^N k_i (X_i - \mu)</math>
then ''Y'' has the density <math> \tfrac{1}{s} f(y / s; \alpha, \beta, c, 0) </math> with
<math display="block"> s = \left(\sum_{i = 1}^N |k_i|^\alpha \right )^{\frac{1}{\alpha}} </math>

The asymptotic behavior is described, for <math>\alpha < 2</math>, by:<ref name=":0" />
<math display="block"> f(x) \sim \frac{1}{|x|^{1 + \alpha}} \left (c^\alpha (1 + \sgn(x) \beta) \sin \left (\frac{\pi \alpha}{2} \right ) \frac{\Gamma(\alpha + 1) }{\pi} \right ) </math>
where Γ is the [[Gamma function]] (except that when <math>\alpha \geq 1</math> and <math>\beta  = \pm 1</math>, the tail does not vanish to the left or right, resp., of ''μ'', although the above expression is 0). This "[[heavy tail]]" behavior causes the variance of stable distributions to be infinite for all <math>\alpha <2</math>. This property is illustrated in the log–log plots below.

When <math>\alpha = 2</math>, the distribution is Gaussian (see below), with tails asymptotic to exp(−''x''<sup>2</sup>/4''c''<sup>2</sup>)/(2''c''{{radic|{{pi}}}}).