The Levy alpha-Stable Distribution

gsl_ran_levy(c, 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,

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

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 \(0 < \alpha \leq 2\).