Stein characterizations for linear combinations of gamma random variables

TL;DR

This paper introduces a new method for deriving Stein operators for certain random variables, focusing on linear combinations of gamma variables and their connections to Malliavin calculus, with applications to McKay Type I distributions.

Contribution

It presents a novel, explicit mechanism for Stein operator derivation for variables with characteristic functions satisfying a simple ODE, especially for linear combinations of gamma variables.

Findings

Derived Stein operators for gamma combinations

Connected Stein methods with Malliavin calculus in Wiener chaos

Applied results to McKay Type I distributions

Abstract

In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study random variables which can be represented as linear combinations of (non necessarily independent) gamma distributed random variables. The connection with Malliavin calculus for random variables in the second Wiener chaos is detailed. An application to McKay Type I random variables is also outlined.