# Levy stable distributions via associated integral transform

Source: [Levy stable distributions via associated integral transform](<https://www.sota2.com/research/paper/68fa30251e209131ce872bd1>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-27T03:31:03.667Z

Authors: K. Gorska, K. A. Penson

Published: 2012-02-08T18:24:20.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1202.1789>)

[PDF](<https://arxiv.org/pdf/1202.1789.pdf>)