The Levy skew alphaStable Distribution¶

gsl_ran_levy_skew
(c, alpha, beta)¶ This function returns a random variate from the Levy skew stable distribution with scale
c
, exponentalpha
and skewness parameterbeta
. The skewness parameter must lie in the range [1,1]. The Levy skew stable probability distribution is defined by a Fourier transform,\[p(x) = {1 \over 2 \pi} \int_{\infty}^{+\infty} \exp(it x  c t^\alpha (1i \beta \operatorname{sign}(t) \tan(\pi \alpha/2))) dt\]When \(\alpha = 1\) the term \(\tan(\pi \alpha/2)\) is replaced by \((2/\pi)\logt\). 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 alphastable distributions have the property that if \(N\) alphastable 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 alphastable variate, \(p(N^{1/\alpha} c, \alpha, \beta)\).