The variance-gamma product distribution

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.