Levy stable two-sided distributions: exact and explicit densities for asymmetric case

TL;DR

This paper derives exact, explicit density functions for one-dimensional Levy stable distributions with rational parameters, covering both symmetric and asymmetric cases, and demonstrates their applications through analytical and graphical analysis.

Contribution

It provides new explicit formulas for Levy stable densities with rational parameters, expanding the set of known solutions and enabling better modeling of experimental data.

Findings

Exact density formulas for rational alpha and beta cases

Reproduction of known results and derivation of new examples

Potential applications in modeling experimental and statistical data

Abstract

We study the one-dimensional Levy stable density distributions g(alpha, beta; x) for -infty < x < infty, for rational values of index alpha and the asymmetry parameter beta: alpha = l/k and beta = (l - 2r)/k, where l, k and r are positive integers such that 0 < l/k < 1 for 0 <= r <= l and 1 < l/k <= 2 for l-k <= r <= k. We treat both symmetric (beta = 0) and asymmetric (beta neq 0) cases. We furnish exact and explicit forms of g(alpha, beta; x) in terms of known functions for any admissible values of alpha and beta specified by a triple of integers k, l and r. We reproduce all the previously known exact results and we study analytically and graphically many new examples. We point out instances of experimental and statistical data that could be described by our solutions.