# 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, $$p(N^{1/\alpha} c, \alpha, \beta)$$.