Stable Distribution - Special Cases

Special Cases

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.

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).