Stable distribution

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Not to be confused with Stationary distribution.

Stable
Probability density function Symmetric α-stable distributions with unit scale factor Skewed centered stable distributions with unit scale factor
Cumulative distribution function CDFs for symmetric α-stable distributions CDFs for skewed centered stable distributions
Parameters	α ∈ (0, 2] — stability parameter β ∈ [−1, 1] — skewness parameter (note that skewness is undefined) c ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R, or x ∈ [μ, +∞) if α < 1 and β = 1, or x ∈ (-∞, μ] if α < 1 and β = −1
pdf	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Mean	μ when α > 1, otherwise undefined
Median	μ when β = 0, otherwise not analytically expressible
Mode	μ when β = 0, otherwise not analytically expressible
Variance	2c² when α = 2, otherwise infinite
Skewness	0 when α = 2, otherwise undefined
Ex. kurtosis	0 when α = 2, otherwise undefined
Entropy	not analytically expressible, except for certain parameter values
MGF	undefined
CF	$\exp \!{\Big [}\;it\mu -\|c\,t\|^{\alpha }\,(1-i\beta \,{\mbox{sgn}}(t)\Phi )\;{\Big ]},$ where $\Phi ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1\\-{\tfrac {2}{\pi }}\log \|t\|&{\text{if }}\alpha =1\end{cases}}$

In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution.

The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically-distributed (iid) random variables. The normal distribution is one 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 towards a normal distribution as the number of variables increases. Without the finite variance assumption the limit may be a stable distribution. Stable distributions that are non-normal are often called stable Paretian distributions,^{[citation needed]} after Vilfredo Pareto.

q-analogs of all symmetric stable distributions have been defined, and these recover the usual symmetric stable distributions in the limit of q → 1.^[1]

Definition

A non-degenerate distribution is a stable distribution if it satisfies the following property:

Let X₁ and X₂ be independent copies of a random variable X. Then X is said to be stable if for any constants a > 0 and b > 0 the random variable aX₁ + bX₂ has the same distribution as cX + d for some constants c > 0 and d. The distribution is said to be strictly stable if this holds with d = 0.^[2]

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 distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters β and α, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures).

Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be. Any probability distribution is determined by its characteristic function φ(t) by:

$f(x)={\frac {1}{2\pi }}\int _{{-\infty }}^{\infty }\varphi (t)e^{{-ixt}}\,dt$

A random variable X is called stable if its characteristic function can be written as^[2]^[3]

$\varphi (t;\mu ,c,\alpha ,\beta )=\exp \left[~it\mu \!-\!|ct|^{\alpha }\,(1\!-\!i\beta \,{\textrm {sgn}}(t)\Phi )~\right]$

where sgn(t) is just the sign of t and Φ is given by

$\Phi =\tan(\pi \alpha /2)\,$

for all α except α = 1 in which case:

$\Phi =-{\frac {2}{\pi }}\log |t|.\,$

μ ∈ R is a shift parameter, β ∈ [−1, 1], called the skewness parameter, is a measure of asymmetry. Notice that in this context the usual skewness is not well defined, as for α < 2 the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment.

In the simplest case β = 0, 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 α < 1 and β = 1, the distribution is supported by [μ, ∞).

The parameter |c| > 0 is a scale factor which is a measure of the width of the distribution and α is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution for α < 2.

Parameterizations

The above definition is only one of the parameterizations in use for stable distributions; it is the most common but is not continuous in the parameters. For example, for the case α = 1 we could replace Φ by:^[2]

$\Phi '=-{\frac {1}{\pi }}\ln \left({\frac {t}{|c|}}\right)$

and μ by

$\mu '=\mu +{\frac {\beta |c|\ln(|c|)}{\pi }}$

This parameterization has the advantage that we may define a standard distribution using

$\tau ={\frac {t}{|c|}}$

and

$y={\frac {x-\mu '}{|c|}}$

The pdf for all α will then have the following standardization property:

$f(x;\mu ',c,\alpha ,\beta )dx=f(y;0,1,\alpha ,\beta )dy\,$

Applications

Stable distributions owe their importance in both theory and practice to the generalization of the central limit theorem to random variables without second (and possibly first) order moments and the accompanying self-similarity of the stable family. It was the seeming departure from normality along with the demand for a self-similar model for financial data (i.e. the shape of the distribution for yearly asset price changes should resemble that of the constituent daily or monthly price changes) that led Benoît Mandelbrot to propose that cotton prices follow an alpha-stable distribution with α equal to 1.7. Lévy distributions are frequently found in analysis of critical behavior and financial data.^[3]

They are also found in spectroscopy as a general expression for a quasistatically pressure-broadened spectral line.^[4]

The statistics of solar flares are described by a non-Gaussian distribution. The solar flare statistics were shown to be describable by a Lévy distribution and it was assumed that intermittent solar flares perturb the intrinsic fluctuations in Earth’s average temperature. The end result of this perturbation is that the statistics of the temperature anomalies inherit the statistical structure that was evident in the intermittency of the solar flare data.^[5]

Lévy distribution of solar flare waiting time events (time between flare events) was demonstrated for CGRO BATSE hard x-ray solar flares December 2001. Analysis of the Lévy statistical signature revealed that two different memory signatures were evident; one related to the solar cycle and the second whose origin appears to be associated with a localized or combination of localized solar active region effects.^[6]

Properties

All stable distributions are infinitely divisible.
With the exception of the normal distribution (α = 2), stable distributions are leptokurtotic and heavy-tailed distributions.
Closure under convolution

Stable distributions are closed under convolution for a fixed value of α. Since convolution is equivalent to multiplication of the Fourier-transformed function, it follows that the product of two stable characteristic functions with the same α will yield another such characteristic function. The product of two stable characteristic functions is given by:

$\exp \left[it\mu _{1}+it\mu _{2}-|c_{1}t|^{\alpha }-|c_{2}t|^{\alpha }+i\beta _{1}|c_{1}t|^{\alpha }{\textrm {sgn}}(t)\Phi +i\beta _{2}|c_{2}t|^{\alpha }\,{\textrm {sgn}}(t)\Phi \right]$

Since Φ is not a function of the μ, c or β variables it follows that these parameters for the convolved function are given by:

$\mu =\mu _{1}+\mu _{2}\,$

$|c|=(|c_{1}|^{\alpha }+|c_{2}|^{\alpha })^{{1/\alpha }}\,$

$\beta ={\frac {\beta _{1}|c_{1}|^{\alpha }+\beta _{2}|c_{2}|^{\alpha }}{|c|^{\alpha }}}$

In each case, it can be shown that the resulting parameters lie within the required intervals for a stable distribution.

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.^[7] If $f(x,\alpha ,\beta ,c,\mu )$ denotes the density of X and Y is the sum of independent copies of X:

$Y=\sum _{{i=1}}^{N}k_{i}(X_{i}-\mu )\,$

then Y has the density $s^{{-1}}f(y/s,\alpha ,\beta ,c,0)$ with

$s=\left(\sum _{{i=1}}^{N}|k_{i}|^{\alpha }\right)^{{1/\alpha }}.\,$

The asymptotic behavior is described, for α< 2, by:^[8]

$f(x)\sim {\frac {c^{\alpha }(1+{\mbox{sgn}}(x)\beta )\sin(\pi \alpha /2)\Gamma (\alpha +1)/\pi }{|x|^{{1+\alpha }}}}$

where Γ is the Gamma function (except that when α < 1 and β = ±1, the tail vanishes to the left or right, resp., of μ). This "heavy tail" behavior causes the variance of stable distributions to be infinite for all α < 2. This property is illustrated in the log-log plots below.

When α = 2, the distribution is Gaussian (see below), with tails asymptotic to exp(−x²/4c²)/(2c√π).

Special cases

Log-log plot of symmetric centered stable distribution PDF's showing the power law behavior for large x. The power law behavior is evidenced by the straight-line appearance of the PDF for large x, with the slope equal to −(α+1). (The only exception is for α = 2, in black, which is a normal distribution.)

Log-log plot of skewed centered stable distribution PDF's showing the power law behavior for large x. Again the slope of the linear portions is equal to −(α+1)

There is no general analytic solution for the form of p(x). There are, however three special cases which can be expressed in terms of elementary functions as can be seen by inspection of the characteristic function.

For α = 2 the distribution reduces to a Gaussian distribution with variance σ² = 2c² and mean μ; the skewness parameter β has no effect.^[2]^[3]

For α = 1 and β = 0 the distribution reduces to a Cauchy distribution with scale parameter c and shift parameter μ.^[3]^[2]
For α =1/2 and β = 1 the distribution reduces to a Lévy distribution with scale parameter c and shift parameter μ.^[4]^[2]

Note that the above three distributions are also connected, in the following way: A standard Cauchy random variable can be viewed as a mixture of Gaussian random variables (all with mean zero), with the variance being drawn from a standard Lévy distribution. And in fact this is a special case of a more general theorem which allows 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 one).

A general closed form expression for stable PDF's with rational values of α is available in terms of Meijer G-functions.^[9] For simple rational numbers, the closed form expression is often in terms of less complicated special functions. Several closed form expressions having rather simple expressions in terms of special functions are available.^[10] In the table below, PDF's expressible by elementary functions are indicated by an E and those that are expressible by special functions are indicated by an s.^[10]

	α
	1/3	1/2	2/3	1	4/3	3/2	2
β=0	s	s	s	E	s	s	E
β=1	s	E	s	s		s	E

Some of the special cases are known by particular names:

For α = 1 and β = 1, the distribution is a Landau distribution which has a specific usage in physics under this name.
For α = 3/2 and β = 0 the distribution reduces to a Holtsmark distribution with scale parameter c and shift parameter μ.^[10]

Also, in the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function δ(x−μ).

A generalized central limit theorem

Another important property of stable distributions is the role that they play in a generalized central limit theorem. The central limit theorem states that the sum of a number of independent and identically distributed (i.i.d.) random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as |x|^−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution $f(x;\alpha ,0,c,0)$ as the number of variables grows. ^[3]

Series representation

The stable distribution can be restated as the real part of a simpler integral:^[4]

$f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left[\int _{0}^{\infty }e^{{it(x-\mu )}}e^{{-(ct)^{\alpha }(1-i\beta \Phi )}}\,dt\right].$

Expressing the second exponential as a Taylor series, we have:

$f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left[\int _{0}^{\infty }e^{{it(x-\mu )}}\sum _{{n=0}}^{\infty }{\frac {(-qt^{\alpha })^{n}}{n!}}\,dt\right]$

where $q=c^{\alpha }(1-i\beta \Phi )$ . Reversing the order of integration and summation, and carrying out the integration yields:

$f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left[\sum _{{n=1}}^{\infty }{\frac {(-q)^{n}}{n!}}\left({\frac {i}{x-\mu }}\right)^{{\alpha n+1}}\Gamma (\alpha n+1)\right]$

which will be valid for x ≠ μ and will converge for appropriate values of the parameters. (Note that the n = 0 term which yields a delta function in x−μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x−μ which is generally less useful.

Notes

↑ Umarov, Sabir; Tsallis, Constantino, Gell-Mann, Murray and Steinberg, Stanly (2010). "Generalization of symmetric α-stable Lévy distributions for q>1". J Math Phys. (American Institute of Physics) 51 (3). arXiv:0911.2009. Bibcode:2010JMP....51c3502U. doi:10.1063/1.3305292. PMC 2869267. PMID 20596232. Retrieved 2011-07-29. Cite uses deprecated parameters (help)
↑ 2.0 2.1 2.2 2.3 2.4 2.5 Nolan (2009)
↑ 3.0 3.1 3.2 3.3 3.4 Voit, Johannes (2003). The Statistical Mechanics of Financial Markets (Texts and Monographs in Physics). Springer-Verlag. ISBN 3-540-00978-7. (Section 5.4.3)
↑ 4.0 4.1 4.2 Peach, G. (1981). "Theory of the pressure broadening and shift of spectral lines". Advances in Physics 30 (3): 367–474. Bibcode:1981AdPhy..30..367P. doi:10.1080/00018738100101467. (Section 4.5)
↑ Scafetta, N., Bruce, J.W., Is climate sensitive to solar variability? Physics Today, 60, 50-51 (2008) .
↑ Leddon, D., A statistical Study of Hard X-Ray Solar Flares
↑ Nolan (2009; Theorem 1.9)
↑ Nolan (2009; Theorem 1.12)
↑ Zolotarev, V.M. (1995). "On Representation of Densities of Stable Laws by Special Functions". Theory Probab. Appl. (SIAM) 39 (2): 354–362. doi:10.1137/1139025. Retrieved 2011-08-15.
↑ 10.0 10.1 10.2 Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.

References

Nolan, John P. (2009). "Stable Distributions: Models for Heavy Tailed Data" (PDF). Retrieved 2009-02-21.

External links

PlanetMath stable random variable article
John P. Nolan page on stable distributions
Applications of stable laws in finance.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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