Stable distribution

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 $\Phi = \begin{cases} \tan\tfrac{\pi\alpha}{2} & \text{if }\alpha \ne 1 \\ -\tfrac{2}{\pi}\log|t| & \text{if }\alpha = 1 \end{cases}$

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, after Vilfredo Pareto.

Umarov, Tsallis, Gell-Mann and Steinberg have defined q-analogs of all symmetric stable distributions which recover the usual symmetric stable distributions in the limit of q->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. Random variable X is said to be stable if for any constants a and b the random variable aX₁ + bX₂ has the same distribution as cX + d with some constants c and d. The distribution is said to be strictly stable if this holds with d = 0 (Nolan 2009).

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 is given by (see Nolan (2009) and Voit (2003, § 5.4.3))

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

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:

$\Phi=-(2/\pi)\log|t|.\,$

$\mu\in\mathbb{R}$ is a shift parameter, $\beta\in[-1,1]$, 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.

When α<1, β=1 and μ=0, the distribution is supported by $[0,\infty).$

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: (Nolan 2009)

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

and $\mu\,$ 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 (Voit 2003, § 5.4.3).

They are also found in spectroscopy as a general expression for a quasistatically pressure-broadened spectral line (Peach 1981, § 4.5).

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.

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.

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 $\Phi\,$ is not a function of the $\mu,\,c$ or $\beta\,$ 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.(Nolan 2009, Theorem 1.9) If f(x,α,β,c,μ) denotes the density of X and $Y = \sum_{i=1}^N k_i (X_i-\mu)\,$ (sum of independent copies of X) then Y has the density s ^{− 1}f(y / s,α,β,c,0) with

$s=\left(\sum_{i=1}^N |k_i|^\alpha\right)^{1/\alpha}.\,$

The asymptotic behavior is described, for α < 2, by: (Nolan 2009, Theorem 1.12)

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

where Γ is the Gamma function (except that when α < 1 and β = 1 or −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 (Nolan 2009) (Voit 2003, § 5.4.3).
• For α = 1 and β = 0 the distribution reduces to a Cauchy distribution with scale parameter c and shift parameter μ (Voit 2003, § 5.4.3) (Nolan 2009).
• For α = 1 / 2 and β = 1 the distribution reduces to a Lévy distribution with scale parameter c and shift parameter μ. (Peach 1981, § 4.5)(Nolan 2009)

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 has been given by Zolotarev in terms of Meijer G-functions. For simple rational numbers, the closed form expression is often in terms of less complicated special functions. Lee(Lee 2010, § 2.4) has listed a number of closed form expressions having rather simple expressions in terms of special functions. In the table below, PDF's expressible by elementary functions are indicated by an E and those given by Lee that are expressible by special functions are indicated by an s.

	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

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.(Lee 2010, § 2.4)

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 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)

Series representation

The stable distribution can be restated as the real part of a simpler integral:(Peach 1981, § 4.5)

$f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \int_0^\infty e^{it(x-\mu)}e^{-(ct)^\alpha(1-i\beta\Phi)}\,dt\right].$

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^α(1 − iβΦ). 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 $x\ne\mu$ 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.

See also

• Lévy flight
• Lévy process
• Lévy distribution
• Fractional quantum mechanics
• Other "power law" distributions
  • Cauchy distribution (a special case of a stable distribution)
  • Pareto distribution
  • Zeta distribution
  • Zipf distribution
  • Zipf–Mandelbrot distribution
• Stable and tempered stable distributions with volatility clustering – financial applications
• Multivariate stable distribution

References

• Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley. ISBN 0-471-25709-5
• Gnedenko, B. V.; Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley. 
• Ibragimov, I.; Linnik, Yu (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands. 
• Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham. http://etheses.nottingham.ac.uk/1194/. 
• Matsui, M.; Takemura, A.. "Some improvements in numerical evaluation of symmetric stable density and its derivatives" (PDF). CIRGE Discussion paper. http://www.e.u-tokyo.ac.jp/cirje/research/dp/2004/2004cf292.pdf. Retrieved July 13, 2005. 
• Nolan, John P. (2009). "Stable Distributions: Models for Heavy Tailed Data" (PDF). http://academic2.american.edu/~jpnolan/stable/chap1.pdf. Retrieved 2009-02-21. 
• 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. http://journalsonline.tandf.co.uk/openurl.asp?genre=article&eissn=1460-6976&volume=30&issue=3&spage=367. 
• Rachev, S.; Mittnik, S. (2000). "Stable Paretian Models in Finance". Wiley. ISBN 978-0471953142. 
• Samorodnitsky, G.; Taqqu, M. (1994). "Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Stochastic Modeling Series)". Chapman and Hall/CRC. ISBN 978-0412051715. 
• Voit, Johannes (2003). The Statistical Mechanics of Financial Markets (Texts and Monographs in Physics). Springer-Verlag. ISBN 3-540-00978-7. 
• Zolotarev, V.M. (1986). One-dimensional Stable Distributions. American Mathematical Society. 

External links

• PlanetMath stable random variable article
• John P. Nolan page on stable distributions
• stable distributions in GNU Scientific Library — Reference Manual
• Applications of stable laws in finance.
• fBasics R package with functions to compute stable density, distribution function, quantile function and generate random variates.
• STBL MATLAB package which includes functions to compute stable densities, CDFs and inverse CDFs. Also can fit stable distributions to data, and generate stable random variables.
• StableDistribution is fully supported in Mathematica since version 8.