1202.1789 (K. Gorska et al.)

Levy stable distributions via associated integral transform    [PDF]

K. Gorska, K. A. Penson

We present a method of generation of exact and explicit forms of one-sided,
heavy-tailed Levy stable probability distributions g_{\alpha}(x), 0 \leq x <
\infty, 0 < \alpha < 1. We demonstrate that the knowledge of one such a
distribution g_{\alpha}(x) suffices to obtain exactly g_{\alpha^{p}}(x), p=2,
3,... Similarly, from known g_{\alpha}(x) and g_{\beta}(x), 0 < \alpha, \beta <
1, we obtain g_{\alpha \beta}(x). The method is based on the construction of
the integral operator, called Levy transform, which implements the above
operations. For \alpha rational, \alpha = l/k with l < k, we reproduce in this
manner many of the recently obtained exact results for g_{l/k}(x). This
approach can be also recast as an application of the Efros theorem for
generalized Laplace convolutions. It relies solely on efficient definite
integration.

View original: http://arxiv.org/abs/1202.1789