Polynomial Stein operators: a noncommutative algebra perspective
Ehsan Azmoodeh, Dario Gasbarra, Robert E. Gaunt
TL;DR
The paper bridges Stein's method with noncommutative algebra by embedding polynomial Stein operators into the first Weyl algebra, enabling a rigorous algebraic analysis of their classes. It completely characterises the Gaussian case, showing PSO$(N)$ is the principal right ideal generated by $G=\partial - x$ with an explicit basis $S(k,t)=H_k\partial^t - H_{k+t}$, and it links this structure to holonomic function theory via Ann$_{A_1}(\varphi_X)$ and Stafford's two-generator results. It further explores higher-order Gaussian Stein operators, showing that while first-order operators are characterising, higher-order ones may fail to be so without additional distributional assumptions, and demonstrates a general intersection phenomenon: holonomic densities or characteristic functions lead to nontrivial intersections of PSO classes. Together, these results reveal deep algebraic reasons for the versatility and limitations of Stein operators and open avenues for applying D-module theory to probabilistic approximation. The work thus provides a new, rigorous framework for selecting and understanding polynomial Stein operators in Gaussian and non-Gaussian settings, with potential implications for Malliavin-Stein methods and holonomic function theory in probability.
Abstract
In this paper, we make a novel connection between Stein's method and noncommutative algebra by viewing polynomial Stein operators (Stein operators with polynomial coefficients) as elements of the first Weyl algebra. Through this connection we study the algebraic structure of classes of polynomial Stein operators. In the case of the standard Gaussian distribution, we provide a complete description of the corresponding class of polynomial Stein operators by (i) identifying it as a vector space over $\mathbb{R}$ with an explicit given basis and (ii) by showing that this class is a principal right ideal of the first Weyl algebra generated by the classical Gaussian Stein operator $\partial -x$, with $\partial$ denoting the usual differential operator. We also study the characterising property of polynomial Stein operators for the standard Gaussian distribution, and give examples of general classes of polynomial Stein operators that are characterising, as well as classes that are not characterising unless additional distributional assumptions are made. By appealing to a standard property of Weyl algebras, we shown that the non-characterising property possessed by a wide class of polynomial Stein operators for the standard Gaussian distribution is a consequence of a general result that is perhaps surprising from a probabilistic perspective: the intersection between the class of polynomial Stein operators for any two target distributions with holonomic densities or holonomic characteristic functions is non-trivial.