Stable distribution

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and 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, the first mathematician to have studied it.
Of the four parameters defining the family, most attention has been focused on the stability parameter, . Stable distributions have, with the upper bound corresponding to the normal distribution, and to the Cauchy distribution. The distributions have undefined variance for, and undefined mean for.
The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed 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. Mandelbrot referred to such distributions as "stable Paretian distributions", after Vilfredo Pareto. In particular, he referred to those maximally skewed in the positive direction with as "Pareto–Lévy distributions", which he regarded as better descriptions of stock and commodity prices than normal distributions.

Definition

A non-degenerate distribution is a stable distribution if it satisfies the following property:
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 and, roughly corresponding to measures of asymmetry and concentration, respectively.
The characteristic function of a probability distribution with density function is the Fourier transform of The density function is then the inverse Fourier transform of the characteristic function:
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
where is just the sign of and
μ ∈ R is a shift parameter,, called the skewness parameter, is a measure of asymmetry. Notice that in this context the usual skewness is not well defined, as for the distribution does not admit 2nd or higher 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 and, but possibly different values of μ and c.
Not every function is the characteristic function of a legitimate probability distribution, 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, the characteristic function is just a stretched exponential function; the distribution is symmetric about μ and is referred to as a symmetric alpha-stable distribution, often abbreviated SαS.
When and, the distribution is supported on independent of. In the first parametrization, if the [mean">Independence (probability theory)">independent of. In the first parametrization, if the [mean exists then it is equal to μ, whereas in the second parametrization when the mean exists it is equal to

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 density function. If denotes the density of X and Y is the sum of independent copies of X:
then Y has the density with
The asymptotic behavior is described, for, by:
where Γ is the Gamma function. This "heavy tail" behavior causes the variance of stable distributions to be infinite for all. This property is illustrated in the log–log plots below.
When, the distribution is Gaussian, with tails asymptotic to exp/.

Properties

Stable distributions are infinitely divisible.
Stable distributions are leptokurtotic and heavy-tailed distributions, with the exception of the normal distribution.
Stable distributions are closed under convolution.

Stable distributions are closed under convolution for a fixed value of. Since convolution is equivalent to multiplication of the Fourier-transformed function, it follows that the product of two stable characteristic functions with the same will yield another such characteristic function. The product of two stable characteristic functions is given by:
Since is not a function of the μ, c or variables it follows that these parameters for the convolved function are given by:
In each case, it can be shown that the resulting parameters lie within the required intervals for a stable distribution.

The Generalized Central Limit Theorem

The Generalized Central Limit Theorem was an effort of multiple mathematicians over the period from 1920 to 1937.
The first published complete proof of the GCLT was in 1937 by Paul Lévy.
An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.
The statement of the GCLT is as follows:
In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a stable distribution.

Special cases

[Image:Levy LdistributionPDF.png|325px|thumb|Log-log plot of symmetric centered stable distribution PDFs 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. (The only exception is for, in black, which is a normal distribution.)]
[Image:Levyskew LdistributionPDF.png|325px|thumb|Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x. Again the slope of the linear portions is equal to ]
There is no general analytic solution for the form of f. 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 the distribution reduces to a Gaussian distribution with variance σ² = 2c² and mean μ; the skewness parameter has no effect.
For and the distribution reduces to a Cauchy distribution with scale parameter c and shift parameter μ.
For and the distribution reduces to a Lévy distribution with scale parameter c and shift parameter μ.

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, 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.
A general closed form expression for stable PDFs with rational values of is available in terms of Meijer G-functions. Fox H-Functions can also be used to express the stable probability density functions. 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. In the table below, PDFs expressible by elementary functions are indicated by an E and those that are expressible by special functions are indicated by an s.
Some of the special cases are known by particular names:

For and, the distribution is a Landau distribution which has a specific usage in physics under this name.
For and the distribution reduces to a Holtsmark distribution with scale parameter c and shift parameter μ.

Also, in the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function.

Series representation

The stable distribution can be restated as the real part of a simpler integral:
Expressing the second exponential as a Taylor series, this leads to:
where. Reversing the order of integration and summation, and carrying out the integration yields:
which will be valid for x ≠ μ and will converge for appropriate values of the parameters. Expressing the first exponential as a series will yield another series in positive powers of x − μ which is generally less useful.
For one-sided stable distribution, the above series expansion needs to be modified, since and. There is no real part to sum. Instead, the integral of the characteristic function should be carried out on the negative axis, which yields:

Parameter estimation

In addition to the existing tests for normality and subsequent parameter estimation, a general method which relies on the quantiles was developed by McCulloch and works for both symmetric and skew stable distributions and stability parameter.

Simulation of stable variates

There are no analytic expressions for the inverse nor the CDF itself, so the inversion method cannot be used to generate stable-distributed variates. Other standard approaches like the rejection method would require tedious computations. An elegant and efficient solution was proposed by Chambers, Mallows and Stuck, who noticed that a certain integral formula yielded the following algorithm:

generate a random variable uniformly distributed on and an independent exponential random variable with mean 1;
for compute:
for compute: where

This algorithm yields a random variable. For a detailed proof see.
To simulate a stable random variable for all admissible values of the parameters,, and use the following property: If then
is. For the CMS method reduces to the well known Box-Muller transform for generating Gaussian random variables. While other approaches have been proposed in the literature, including application of Bergström and LePage series expansions, the CMS method is regarded as the fastest and the most accurate.

Applications

Stable distributions owe their importance in both theory and practice to the generalization of the central limit theorem to random variables without second 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 that led Benoît Mandelbrot to propose that cotton prices follow an alpha-stable distribution with equal to 1.7. Lévy distributions are frequently found in analysis of critical behavior and financial data.
They are also found in spectroscopy as a general expression for a quasistatically pressure broadened spectral line.
The Lévy distribution of solar flare waiting time events was demonstrated for CGRO BATSE hard x-ray solar flares in 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.

Other analytic cases

A number of cases of analytically expressible stable distributions are known. Let the stable distribution be expressed by, then:

The Cauchy Distribution is given by
The Lévy distribution is given by
The Normal distribution is given by
Let be a Lommel function, then:
Let and denote the Fresnel integrals, then:
Let be the modified Bessel function of the second kind, then:
Let denote the hypergeometric functions, then: with the latter being the Holtsmark distribution.
Let be a Whittaker function, then:

Software implementations

The STABLE program for Windows is available from John Nolan's stable webpage: http://www.robustanalysis.com/public/stable.html. It calculates the density, cumulative distribution function and quantiles for a general stable distribution, and performs maximum likelihood estimation of stable parameters and some exploratory data analysis techniques for assessing the fit of a data set.
The GNU Scientific Library which is written in C has a package randist, which includes among the Gaussian and Cauchy distributions also an implementation of the Levy alpha-stable distribution, both with and without a skew parameter.
is a C implementation for the Stable distribution pdf, cdf, random number, quantile and fitting functions.
R Package by Diethelm Wuertz, Martin Maechler and Rmetrics core team members. Computes stable density, probability, quantiles, and random numbers.
Python implementation is located in in the SciPy package.
Julia provides package which has methods of generation, fitting, probability density, cumulative distribution function, characteristic and moment generating functions, quantile and related functions, convolution and affine transformations of stable distributions. It uses modernised algorithms improved by John P. Nolan.