The Levy skew alpha-Stable Distribution

gsl_ran_levy_skew(c, alpha, 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 [-1,1]. The Levy skew stable probability distribution is defined by a Fourier transform,

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

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

The algorithm only works for \(0 < \alpha \leq 2\).

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