论文标题
通过使用分化技术的集成来扩展Stein的引理的扩展
Extension of Stein's lemma derived by using an integration by differentiation technique
论文作者
论文摘要
我们将Stein的引理扩展到平均值，该平均值明确包含电源的高斯随机变量。我们为斯坦因引理的这一扩展提供了两个证据，其中首先是数学诱导的严格证明。替代性的第二个证明是一个建设性的形式推导，其中我们表达了平均值，而不是作为积分，而是通过高斯瞬间产生函数定义的假差异操作员的作用。在扩展的Stein的引理中，出现了概率的Hermite多项式系数的绝对值，从而揭示了Hermite多项式和正态分布之间的另一个联系。
We extend Stein's lemma for averages that explicitly contain the Gaussian random variable at a power. We present two proofs for this extension of Stein's lemma, with the first being a rigorous proof by mathematical induction. The alternative, second proof is a constructive formal derivation in which we express the average not as an integral, but as the action of a pseudodifferential operator defined via the Gaussian moment-generating function. In extended Stein's lemma, the absolute values of the coefficients of the probabilist's Hermite polynomials appear, revealing yet another link between Hermite polynomials and normal distribution.
