多项式 Stein 算子：非交换代数视角
Polynomial Stein operators: a noncommutative algebra perspective
AI总结：
本文将多项式 Stein 算子视为第一 Weyl 代数元素，研究其代数结构，完整刻画了标准高斯分布对应的算子类，并揭示了任意两个 holonomic 目标分布的多项式 Stein 算子类交集非平凡的普遍性质。
摘要：
在本文中，我们通过将多项式 Stein 算子（具有多项式系数的 Stein 算子）视为第一 Weyl 代数的元素，在 Stein 方法和非交换代数之间建立了一种新颖的联系。通过这种联系，我们研究了几类多项式 Stein 算子的代数结构。在标准高斯分布的情况下，我们通过以下方式提供了对应多项式 Stein 算子类的完整描述：(i) 将其识别为 $\mathbb{R}$ 上的向量空间，并给出了明确的基；(ii) 证明该类是由经典高斯 Stein 算子 $\partial -x$ 生成的第一 Weyl 代数的主右理想，其中 $\partial$ 表示通常的微分算子。我们还研究了标准高斯分布的多项式 Stein 算子的刻画性质，并给出了具有刻画性质的广义多项式 Stein 算子类的例子，以及除非做出额外分布假设否则不具有刻画性质的算子类的例子。通过应用 Weyl 代数的标准性质，我们证明了标准高斯分布的一大类多项式 Stein 算子所具有的非刻画性质是一个一般性结果的推论，该结果从概率论角度来看可能令人惊讶：对于具有 holonomic 密度或 holonomic 特征函数的任意两个目标分布，其多项式 Stein 算子类之间的交集是非平凡的。
英文摘要：
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.
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