The variance-gamma product distribution

TL;DR

This paper derives the exact probability density function for the product of multiple independent variance-gamma variables and extends these results to related distributions, providing formulas for density, CDF, and characteristic functions.

Contribution

It introduces a comprehensive derivation of the product distribution of variance-gamma variables and related distributions, including asymptotic approximations and closed-form formulas.

Findings

Exact density function for the product of variance-gamma variables

Formulas for the CDF and characteristic function of the product

Asymptotic approximations for tail probabilities and quantiles

Abstract

We derive the exact probability density function of the product of $N$ independent variance-gamma random variables with zero location parameter. We then apply this formula to derive formulas for the cumulative distribution function and characteristic function, as well as asymptotic approximations for the density, tail probabilities and quantile function. From our general results, we deduce closed-form formulas for the density, cumulative distribution function and characteristic function of the product of $N$ independent asymmetric Laplace random variables and mixed products of independent Laplace and centred normal random variables.