On algebraic Stein operators for Gaussian polynomials

Abstract

The first essential ingredient to build up Stein’s method for a continuous target distribution is to identify a so-called Stein operator, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of algebraic Stein operators (see Definition 3.4), and provide a novel algebraic method to find all the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form Y=h(X), where X=(X1,…,Xd) has i.i.d. standard Gaussian components and h∈K[X] is a polynomial with coefficients in the ring K. Our approach links the existence of an algebraic Stein operator with null controllability of a certain linear discrete system. A MATLAB code checks the null controllability up to a given finite time T (the order of the differential operator), and provides all null control sequences (polynomial coefficients of the differential operator) up to a given maximum degree m. This is the first paper that connects Stein’s method with computational algebra to find Stein operators for highly complex probability distributions, such as H20(X1), where Hp is the p-th Hermite polynomial. Some examples of Stein operators for Hp(X1), p=3,4,5,6, are gathered in the Appendix and many other examples are given in the Supplementary Information.