Levy stable distributions via associated integral transform

TL;DR

The paper introduces a novel integral transform method to generate explicit forms of Levy stable distributions, enabling exact calculations for various parameters and simplifying the derivation process.

Contribution

It develops the Levy transform, an integral operator that systematically produces Levy stable distributions for different parameters, including rational exponents, using only definite integrals.

Findings

Derived explicit forms for Levy distributions with rational stability parameters.

Established a method to generate distributions for composite parameters from known cases.

Connected the approach to Efros theorem for generalized Laplace convolutions.

Abstract

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.