Lévy stable distributions via associated integral transform

Abstract

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Lévy stable probability distributions gα(x), 0 ⩽ x < ∞, 0 < α < 1. We demonstrate that the knowledge of one such a distribution gα(x) suffices to obtain exactly \documentclass[12pt]{minimal}\begin{document}$g_{\alpha ^{p}}(x)$\end{document}gαp(x), p = 2, 3, … . Similarly, from known gα(x) and gβ(x), 0 < α, β < 1, we obtain gαβ(x). The method is based on the construction of the integral operator, called Lévy transform, which implements the above operations. For α rational, α = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for gl/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.