Stein operators for variables form the third and fourth Wiener chaoses

Let Z be a standard normal random variable and let Hn denote the nth Hermite polynomial. In this note, we obtain Stein equations for the random variables H3(Z) and H4(Z), which represent 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 aZ2+bZ+c, a,b,c∈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 Hn(Z), n≥5, is more challenging.