Levy stable distributions via associated integral transform

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g(alpha)(x), 0 <= x < infinity, 0 < alpha < 1. We demonstrate that the knowledge of one such a distribution g a ( 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 a 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. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4709443]

Agence Nationale de la Recherche (Paris, France)

Agence Nationale de la Recherche (Paris, France) [ANR-08-BLAN-0243-2]

PAN/French National Center for Scientific Research (CNRS) [4339]

PAN/French National Center for Scientific Research (CNRS)

Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP, Brasil) [2010/15698-5]

Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP)Brasil