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Lévy stable two-sided distributions: exact and explicit densities for asymmetric case
1Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Université Pierre et Marie Curie, CNRS UMR 7600, Paris, France. kasia gorska@o2.pl
Summary
This study provides exact solutions for heavy-tailed Lévy stable probability distributions, crucial for modeling complex systems in physics and finance. These new findings expand the applicability of these distributions to previously inaccessible parameter ranges.
Area of Science:
- Probability theory
- Statistical modeling
- Complex systems analysis
Background:
- Heavy-tailed distributions, specifically Lévy stable distributions (g(α,β;x)), are vital for understanding phenomena in diverse fields.
- Existing solutions for these distributions are limited, hindering their application in certain complex systems.
Purpose of the Study:
- To derive exact and explicit solutions for Lévy stable probability distributions (g(α,β;x)) across a wider range of parameters.
- To extend the applicability of these distributions to previously inaccessible rational values of the index (α) and asymmetry parameter (β).
Main Methods:
- Analytical derivation of probability distribution functions.
- Ensemble of exact and explicit solution generation.
Main Results:
- A comprehensive set of exact and explicit solutions for g(α,β;x) distributions.
- Reproduction of all previously known solutions.
- Novel solutions for previously inaccessible rational values of α and β.
Conclusions:
- The derived solutions enhance the modeling capabilities for complex systems.
- These findings have potential applications in fields such as amorphous systems physics, econophysics, and hydrology.
- The study provides a valuable resource for researchers analyzing experimental and statistical data exhibiting heavy-tailed behavior.
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