Stable Distribution Computed with the Zolotarev Integral

The graphic shows the value of the cumulative probability distribution function at of a member of a family of standardized stable distributions parametrized by and . There are no explicit formulas for the general case, so calculations are made using numerical integration of the inverse Fourier transform of the characteristic function. The usual representation of this integrand is highly oscillatory. The Zolotarev transform of the integrand is a slowly varying function over a fixed interval. In this representation, total probability is represented as the area of a unit square. The green area is the cumulative probability, the orange area is 1 minus this probability. The integral works only for , but values for can be obtained by the symmetrical properties of stable distributions; at this point you will see the colors in the graph reverse. The function has a closed form at , so this point can always be calculated when . When , the integral formula is different, but does not exist when .
is the shape parameter; when , the distribution is a Normal distribution with and ; has a range of ; in the Demonstration it is supported down to 0.1.
is the skewness parameter and falls in the range .
At and , the distribution is the Cauchy distribution, which cannot be calculated by this transformation.
The number below the graphic is probability (the green area) calculated by numerical integration.
This Demonstration was created as a first step in developing faster numerical methods for calculating stable distributions, perhaps by methods similar to those in plotting graphics. The idea of the graphic is to view the area as a Lebesgue integral: a set of points with each point reflecting a measure of area.

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DETAILS

The Zolotarev transformation of the stable integral is described in detail by Nolan.
J. P. Nolan, "Numerical Calculation of Stable Densities and Distribution Functions," Stochastic Models, 13(4), 1997 pp. 759–774.
A derivation of the function is presented on a web page with a Mathematica notebook at this link.
For many Mathematica notebooks and software for stable distributions visit mathestate.
John Nolan's stable page has a lot of information on stable distributions. There is also a MathLink program for stable distributions.
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