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

TL;DR

This paper extends Stein's method to derive operators and closed-form distributional formulas for products of independent beta, gamma, and normal variables, providing new tools for their probabilistic analysis.

Contribution

It introduces Stein operators for mixed products of beta, gamma, and normal distributions, including explicit formulas for their density and characteristic functions.

Findings

Derived Stein operators for mixed products of distributions

Obtained closed-form density and characteristic functions using Meijer G-functions

Unified classical distributions within a general framework

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 operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions 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.