On Distribution of Product of Stable Laws

TL;DR

This paper derives the probability distribution of the product of two independent stable law variables, providing series and asymptotic representations, and links these to Fox's H-functions for better analytical understanding.

Contribution

It introduces explicit density representations for the product of stable laws using power series and asymptotic expansions, and connects these to Fox's H-functions.

Findings

Derived the density of the product of two stable laws.

Expressed densities using power series and asymptotic expansions.

Connected stable law densities to Fox's H-functions.

Abstract

We derive the probability distribution of product of two independent random variables, each distributed according the one-dimensional stable law. We represent the density by its power series and its asymptotic expansions. As Fox's H-functions with a particular choice of parameters well describe the densities of stable laws, we discuss the choice of parameters for Fox's H-function such that it matches the derived densities. As a consequence, we give representations of these particular Fox's H-functions in terms of its power series.