The Levy skew alpha-Stable Distribution

Random: double gsl_ran_levy_skew (const gsl_rng * r, double c, double alpha, double beta)

This function returns a random variate from the Levy skew stable distribution with scale c, exponent alpha and skewness parameter beta. The skewness parameter must lie in the range @math{[-1,1]}. The Levy skew stable probability distribution is defined by a fourier transform,

When @math{\alpha = 1} the term @math{\tan(\pi \alpha/2)} is replaced by @math{-(2/\pi)\log|t|}. There is no explicit solution for the form of @math{p(x)} and the library does not define a corresponding pdf function. For @math{\alpha = 2} the distribution reduces to a Gaussian distribution with @c{$\sigma = \sqrt{2} c$} @math{\sigma = \sqrt{2} c} and the skewness parameter has no effect. For @math{\alpha < 1} the tails of the distribution become extremely wide. The symmetric distribution corresponds to @math{\beta = 0}.

The algorithm only works for @c{$0 < \alpha \le 2$} @math{0 < alpha <= 2}.

The Levy alpha-stable distributions have the property that if @math{N} alpha-stable variates are drawn from the distribution @math{p(c, \alpha, \beta)} then the sum @math{Y = X_1 + X_2 + \dots + X_N} will also be distributed as an alpha-stable variate, @math{p(N^(1/\alpha) c, \alpha, \beta)}.