Stable distribution: Information from Answers.com

Not to be confused with Stationary distribution.

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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:	$\alpha\in (0,2]\,$ exponent $\beta\in [-1,1]\,$ called skewness parameter (note that skewness is undefined) $c\in [0,\infty)\,$ scale $\mu \in (-\infty,\infty)\,$ location
support:	$x \in (-\infty, +\infty)\!$ or a half-line if $\| β \| = 1$
pdf:	usually not analytically expressible (see text)
cdf:	usually not analytically expressible (see text)
mean:	undefined when α ≤ 1, otherwise μ
median:	usually not analytically expressible (see text). Equal to μ when β = 0
mode:	usually not analytically expressible. Equal to μ when β = 0
variance:	infinite except when α = 2, when it is 2c²
skewness:	undefined except when α = 2, when it is 0
kurtosis:	undefined except when α = 2, when it is 0
entropy:	not analytically expressible (see text)
mgf:	undefined
cf:	$\exp\left[~it\mu - \|c t\|^\alpha\,(1-i \beta\,\mbox{sgn}(t)\Phi)~\right]$ $\Phi=\tan(\pi \alpha/2)\,$ for $\alpha \ne 1\,$ $\Phi=-(2/\pi)\log\|t\|\,$ for $\alpha = 1\,$

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.

1 Definition
2 Applications
3 Properties
4 Other definitions of stability
5 The distribution
6 Special cases
7 The generalized central limit theorem
8 Series representation
9 See also
10 External links
11 References

Definition

The stable distributions are defined by the following property

if X₁ and X₂ are independent copies of a stable random variable X, then 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 $\varphi(t).$ A random variable X is called stable if its characteristic function is given by (Nolan 2009)(Voit 2003 § 5.4.3)

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

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 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. Parameters are not completely independent, for example β=1 is possible only when $\alpha\in(0,1).$

Note that this is only one of the parameterizations in use for stable distributions; it is the most common but is not continuous in the parameters.

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).

Properties

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

Other definitions of stability

Below we give frequently used equivalent definitions of stability (Nolan 2009),(Voit 2003 § 5.4.3).

A random variable X is called stable if for n independent copies $X i$ of X there exists a constant d such that

$X_1+X_2+\ldots+X_n \stackrel{d}{=} n^{1/\alpha} X+d.$

(equality of distributions).

The distribution

A stable distribution is therefore specified by the above four parameters. It can be shown that any stable distribution has continuous (even smooth) density function. 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, Theorem 1.12)

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

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 Lévy 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 analytically expressed as can be seen by inspection of the characteristic function.

For α = 2 the distribution reduces to a Gaussian distribution with variance $σ 2 = 2 c 2$ and mean $μ$ and 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 unity).

Other special cases are:

In the limit as $c$ approaches zero or as $α$ approaches zero the distribution will approach a Dirac delta function $δ(x - μ)$ .
For $α = 1$ and $β = 1$ ,the distribution is a Landau distribution which has a specific usage in physics under this name.

The generalized central limit theorem

Another important property of Lévy 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 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 $1 / | x | α + 1$ (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 $q = c α (1 - i βΦ)$ . 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.

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.

References

Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley. ISBN 0-471-25709-5
B. V. Gnedenko and A. N. Kolmogorov (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.
I. Ibragimov, Yu. Linnik (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands.
M. Matsui and A. Takemura. "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.
John P. Nolan (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. doi:10.1080/00018738100101467. http://journalsonline.tandf.co.uk/openurl.asp?genre=article&eissn=1460-6976&volume=30&issue=3&spage=367.
Johannes Voit (2003). The Statistical Mechanics of Financial Markets (Texts and Monographs in Physics). Springer-Verlag. ISBN 3-540-00978-7.
V.M. Zolotarev (1986). One-dimensional Stable Distributions. American Mathematical Society.

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway–Maxwell–Poisson · compound Poisson · discrete phase-type · 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]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell–Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · 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 · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

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Stable distribution

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