论文标题
多项式Stein操作员：非共同代数观点
Polynomial Stein operators: a noncommutative algebra perspective
论文作者
论文摘要
在本文中，我们通过将Stein的方法与非共同代数之间建立了新的联系，通过将多项式Stein运算符（带有多项式系数的Stein运算符）作为第一个Weyl代数的元素。通过这种联系，我们研究了多项式Stein运算符类别的代数结构。在标准高斯分布的情况下，我们通过（i）通过（i）通过（i）将其识别为$ \ mathbb {r} $的矢量空间的相应类别类别，并以明确的基础和（ii）表明该类是$ $ n e $ necy $ $ $ $ $ $ $ $ $ $ $ s $ progiant $ s $ sepy $ s $ s $ s $ s $ s $ s $ n offer $ s $ s $ n offeral $。通常的差分运算符。我们还研究了多项式Stein操作员在标准高斯分布中的特征，并给出了表征的多项式Stein运算符的一般类别的示例，除非做出其他分布假设，否则没有表征的类别。通过吸引Weyl代数的标准特性，我们表明，一类多种多样的Stein操作员所拥有的非特征性特性用于标准高斯分布，这是由于概率的角度来看的一般结果：从概率的角度来看，这可能是一种与任何义务分配者之间的相互作用相互分配，具有任何与任何义务的特征性或自治性良好的特征。
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.
