Closed-form solutions for the Lévy-stable distribution

Karina Arias-Calluari, Fernando Alonso-Marroquin, and Michael S. Harré
Phys. Rev. E 98, 012103 – Published 3 July 2018

Abstract

The Lévy-stable distribution is the attractor of distributions which hold power laws with infinite variance. This distribution has been used in a variety of research areas; for example, in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport. This function does not have an explicit expression and no uniform solution has been proposed yet. This paper presents a uniform analytical approximation for the Lévy-stable distribution based on matching power series expansions. For this solution, the trans-stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the Lévy-stable distribution. Then, the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called “the inner solution” and “the outer solution.” Finally, the results of analytical approximation are compared to the numerical results of the Lévy-stable distribution function, making this uniform solution valid to be applied as an analytical approximation.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
7 More
  • Received 13 December 2017
  • Revised 30 April 2018

DOI:https://doi.org/10.1103/PhysRevE.98.012103

©2018 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Karina Arias-Calluari1,*, Fernando Alonso-Marroquin1,2, and Michael S. Harré3

  • 1School of Civil Engineering, The University of Sydney, Sydney NSW 2006, Australia
  • 2Computational Physics IfB, ETH Zurich, CH-8093 Zurich, Switzerland
  • 3Complex Systems Research Group, The University of Sydney, Sydney NSW 2006, Australia

  • *kari0293@uni.sydney.edu.au

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 98, Iss. 1 — July 2018

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×