# Some new Stein operators for product distributions

## TL;DR

This paper develops a general method for deriving Stein operators for the product of two independent random variables, extending previous work and applying to various distributions including normals and gamma, with insights into their complexity.

## Contribution

It introduces a unified framework for Stein operators of product distributions, covering non-centered normals, gamma, and variance-gamma distributions, expanding prior results.

## Key findings

- Derived Stein operators for products of independent variables.
- Provided a simple derivation of the characteristic function for product of normals.
- Explained the increased complexity of the PDF in non-centered cases.

## Abstract

We provide a general result for finding Stein operators for the product of two independent random variables whose Stein operators satisfy a certain assumption, extending a recent result of Gaunt, Mijoule and Swan \cite{gms18}. This framework applies to non-centered normal and non-centered gamma random variables, as well as a general sub-family of the variance-gamma distributions. Curiously, there is an increase in complexity in the Stein operators for products of independent normals as one moves, for example, from centered to non-centered normals. As applications, we give a simple derivation of the characteristic function of the product of independent normals, and provide insight into why the probability density function of this distribution is much more complicated in the non-centered case than the centered case.