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.

A random variable X is called stable if its characteristic function can be written as[7][9]

, 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.

The parametrization above is easiest to use for theoretical work, but its probability density is not continuous in the parameters at

It can be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function.

This "heavy tail" behavior causes the variance of stable distributions to be infinite for all

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 (GCLT) was an effort of multiple mathematicians (Berstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937.

[14] The first published complete proof (in French) of the GCLT was in 1937 by Paul Lévy.

[15] An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.

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:[7][9][17] 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 (See p. 59 of [18]) 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 PDFs with rational values of

[19] 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.

The stable distribution can be restated as the real part of a simpler integral:[20]

(Note that the n = 0 term which yields a delta function in x − μ has therefore been dropped.)

Instead, the integral of the characteristic function should be carried out on the negative axis, which yields:[21][12]

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

[11][13] Other standard approaches like the rejection method would require tedious computations.

An elegant and efficient solution was proposed by Chambers, Mallows and Stuck (CMS),[23] who noticed that a certain integral formula[24] yielded the following algorithm:[25] This algorithm yields a random variable

[26] To simulate a stable random variable for all admissible values of the parameters

) the CMS method reduces to the well known Box-Muller transform for generating Gaussian random variables.

[27] While other approaches have been proposed in the literature, including application of Bergström[28] and LePage[29] series expansions, the CMS method is regarded as the fastest and the most accurate.

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

[6] Lévy distributions are frequently found in analysis of critical behavior and financial data.

[9][30] They are also found in spectroscopy as a general expression for a quasistatically pressure broadened spectral line.

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.