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19.12 The Levy alpha-Stable Distributions

Function: double gsl_ran_levy (const gsl_rng * r, double c, double alpha)

This function returns a random variate from the Levy symmetric stable distribution with scale c and exponent alpha. The symmetric stable probability distribution is defined by a fourier transform,

p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For \alpha = 1 the distribution reduces to the Cauchy distribution. For \alpha = 2 it is a Gaussian distribution with \sigma = \sqrt{2} c. For \alpha < 1 the tails of the distribution become extremely wide.

The algorithm only works for 0 < alpha <= 2.