One-sided L\'{e}vy stable distributions

TL;DR

This paper introduces new mathematical representations of one-sided Lévy stable distributions for irrational indices, extends existing formulas for rational indices, and proposes novel concepts like Lévy smashing and Lévy-smashed gamma processes.

Contribution

It provides new representations and proofs for Lévy stable distributions with irrational indices and introduces the concepts of Lévy smashing and Lévy-smashed gamma processes.

Findings

New representations for irrational Lévy indices.

Extended formulas for rational Lévy indices.

Introduction of Lévy smashing and Lévy-smashed gamma processes.

Abstract

In this paper, we show new representations of one-sided L\'{e}vy stable distributions for irrational L\'{e}vy indices of the type $\left(\frac{p}{q}\right)^{\frac{l_{2}}{l_{1}}}$ which are not covered in \cite{pg1} : for rational L\'{e}vy indices. Furthermore, other equivalent representations for a distribution of a rational L\'{e}vy index is described. We also give a simplest proof for the formulae which cover the cases for rational L\'{e}vy indices. Finally we introduce the concepts of L\'{e}vy smashing and L\'{e}vy-smashed gamma stochastic processes.