


The Levy skew alpha-Stable Distribution
=======================================

.. function:: gsl_ran_levy_skew(c, alpha, beta)

 This function returns a random variate from the :index:`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,

 .. math::
   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 :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 :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 :math:`0 < \alpha \leq 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)`.
