Editing Stable distribution (section)

{{Short description|Distribution of variables which satisfies a stability property under linear combinations}}
{{Distinguish|Stationary distribution}}

{{Probability distribution
  | name       = Stable
  | type       = continuous
  | pdf_image  = [[Image:Levy distributionPDF.svg|325px|Symmetric stable distributions]]<br /><small>Symmetric <math>\alpha</math>-stable distributions with unit scale factor</small><br />[[Image:Levyskew distributionPDF.svg|325px|Skewed centered stable distributions]]<br /><small>Skewed centered stable distributions with unit scale factor</small>
  | cdf_image  = [[Image:Levy distributionCDF.svg|325px|CDFs for symmetric <math>\alpha</math>'-stable distributions]]<br /><small>CDFs for symmetric <math>\alpha</math>-stable distributions</small> <br />[[Image:Levyskew distributionCDF.svg|325px|CDFs for skewed centered Lévy distributions]]<br /><small>CDFs for skewed centered stable distributions</small>
  | parameters = <math>\alpha \in (0,2]</math> — stability parameter <br>
<math>\beta</math> ∈ [−1, 1] — skewness parameter (note that [[skewness]] is undefined)<br>
''c'' ∈ (0, ∞) — [[scale parameter]] <br>
''μ'' ∈ (−∞, ∞) — [[location parameter]]
  | support    = 
''x'' ∈ [''μ'', +∞) if <math>\alpha < 1</math> and <math>\beta=1</math>

''x'' ∈ (-∞, ''μ''] if <math>\alpha < 1</math> and <math>\beta = -1</math>

''x'' ∈ '''R''' otherwise
  | pdf        = not analytically expressible, except for some parameter values
  | cdf        = not analytically expressible, except for certain parameter values
  | mean       = ''μ'' when <math>\alpha > 1</math>, otherwise undefined
  | median     = ''μ'' when <math>\beta = 0</math>, otherwise not analytically expressible
  | mode       = ''μ'' when <math>\beta = 0</math>, otherwise not analytically expressible
  | variance   = 2''c''<sup>2</sup> when <math>\alpha = 2</math>, otherwise infinite
  | skewness   = 0 when <math>\alpha = 2</math>, otherwise undefined
  | kurtosis   = 0 when <math>\alpha = 2</math>, otherwise undefined
  | entropy    = not analytically expressible, except for certain parameter values
  | mgf        = <math>\exp\!\big(t\mu + c^2t^2\big)</math> when <math>\alpha = 2</math>,<br> <math>\exp\!\big(t\mu - c^\alpha t^\alpha \sec(\pi\alpha/2)\big)</math> when <math>\alpha \neq 1, \beta = -1, t>0</math>,<br> <math>\exp\!\big(t\mu -c2\pi^{-1}t\ln t\big)</math> when <math>\alpha = 1,\beta=-1,t>0</math>,<br> otherwise undefined
  | char       = <math>\exp\!\Big[\; it\mu - |c\,t|^\alpha\,(1-i \beta\sgn(t)\Phi) \;\Big],</math><br>
where <math>\Phi = \begin{cases} \tan\tfrac{\pi\alpha}{2} & \text{if }\alpha \ne 1 \\ -\tfrac{2}{\pi}\log|t| & \text{if }\alpha = 1 \end{cases}</math>}}

In [[probability theory]], a [[probability distribution|distribution]] is said to be '''stable''' if a [[linear combination]] of two [[Independence (probability theory)|independent]] [[random variable]]s with this distribution has the same distribution, [[up to]] [[location parameter|location]] and [[scale parameter|scale]] parameters. A random variable is said to be '''stable''' if its distribution is stable. The stable distribution family is also sometimes referred to as the '''Lévy alpha-stable distribution''', after [[Paul Lévy (mathematician)|Paul Lévy]], the first mathematician to have studied it.<ref name="BM 1960">{{cite journal |first=B. |last=Mandelbrot |title=The Pareto–Lévy Law and the Distribution of Income |journal=International Economic Review |volume=1 |issue=2 |year=1960 |pages=79–106 |doi=10.2307/2525289 |jstor=2525289 }}</ref><ref>{{cite book |first=Paul |last=Lévy |title=Calcul des probabilités |location=Paris |publisher=Gauthier-Villars |year=1925 |oclc=1417531 }}</ref>

Of the four parameters defining the family, most attention has been focused on the stability parameter, <math>\alpha</math> (see panel). Stable distributions have <math>0 < \alpha \leq 2</math>, with the upper bound corresponding to the [[normal distribution]], and <math>\alpha=1</math> to the [[Cauchy distribution]]. The distributions have undefined [[variance]] for <math>\alpha < 2</math>, and undefined [[mean]] for <math>\alpha \leq 1</math>. The importance of stable probability distributions is that they are "[[attractor]]s" for properly normed sums of independent and identically distributed ([[iid]]) random variables. The normal distribution defines a family of stable distributions. By the classical [[central limit theorem]] the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal.  [[Benoit Mandelbrot|Mandelbrot]] referred to such distributions as "stable Paretian distributions",<ref>{{cite journal |first=B. |last=Mandelbrot |title=Stable Paretian Random Functions and the Multiplicative Variation of Income |journal=Econometrica |volume=29 |issue=4 |pages=517–543 |year=1961 |doi=10.2307/1911802 |jstor=1911802 }}</ref><ref>{{cite journal |first=B. |last=Mandelbrot |title=The Variation of Certain Speculative Prices |journal=The Journal of Business |volume=36 |issue=4 |pages=394–419 |year=1963 |doi=10.1086/294632 |jstor=2350970 }}</ref><ref>{{cite journal |first=Eugene F. |last=Fama |title=Mandelbrot and the Stable Paretian Hypothesis |journal=The Journal of Business |volume=36 |issue=4 |pages=420–429 |year=1963 |doi=10.1086/294633 |jstor=2350971 }}</ref> after [[Vilfredo Pareto]]. In particular, he referred to those maximally skewed in the positive direction with <math>1 < \alpha < 2</math> as "Pareto–Lévy distributions",<ref name="BM 1960"/> which he regarded as better descriptions of stock and commodity prices than normal distributions.<ref name="BM 1963">{{cite journal |last=Mandelbrot |first=B. |title=New methods in statistical economics |journal=[[The Journal of Political Economy]] |volume=71 |issue=5 |pages=421–440 |year=1963 |doi=10.1086/258792 |s2cid=53004476 }}</ref>