Lévy stable distributions via associated integral transform
We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Lévy stable probability distributions g α(x), 0 ⩽ x < ∞, 0 < α < 1. We demonstrate that the knowledge of one such a distribution g α(x) suffices to obtain exactly \documentclass[12pt]{minimal}\begin{doc...
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Veröffentlicht in: | Journal of mathematical physics 2012-05, Vol.53 (5), p.053302-053302-10 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
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Zusammenfassung: | We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Lévy stable probability distributions g
α(x), 0 ⩽ x < ∞, 0 < α < 1. We demonstrate that the knowledge of one such a distribution g
α(x) suffices to obtain exactly
\documentclass[12pt]{minimal}\begin{document}$g_{\alpha ^{p}}(x)$\end{document}
g
α
p
(
x
)
, p = 2, 3, … . Similarly, from known g
α(x) and g
β(x), 0 < α, β < 1, we obtain g
αβ(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 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. |
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ISSN: | 0022-2488 1089-7658 |
DOI: | 10.1063/1.4709443 |