Stein operators for variables form the third and fourth Wiener chaoses

TL;DR

This paper develops Stein equations for specific third and fourth Wiener chaos variables, advancing the methodology for distributional limits in stochastic analysis.

Contribution

It derives new Stein equations for H_3(Z) and H_4(Z), the first such equations for these chaos levels, aiding Stein's method for complex distributions.

Findings

Stein equations for H_3(Z) and H_4(Z) are fifth and third order ODEs.

Established a Stein equation for quadratic forms and non-central chi-square distributions.

Discussed challenges in extending Stein equations to higher Wiener chaoses.

Abstract

Let $Z$ be a standard normal random variable and let $H_n$ denote the $n$-th Hermite polynomial. In this note, we obtain Stein equations for the random variables $H_3(Z)$ and $H_4(Z)$, which represents a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable $aZ^2+bZ+c$, $a,b,c\in\mathbb{R}$, which leads us to a Stein equation for the non-central chi-square distribution. We also provide a discussion as to why obtaining Stein equations for $H_n(Z)$, $n\geq5$, is more challenging.