Products of normal, beta and gamma random variables: Stein operators and distributional theory

Abstract

In this paper, we extend Stein's method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein characterisations for mixed products of these distributions, which include the classical beta, gamma and normal characterisations as special cases. These characterisations, lead us to closed form formulas, involving the Meijer \(G\)-function, for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.