Stein引理的微分积分法推广
Extension of Stein's lemma derived by using an integration by differentiation technique
摘要：
我们扩展斯坦引理的平均值，明确包含高斯随机变量的权力。我们提出了两个证明这个扩展的斯坦引理，第一个是严格的证明数学归纳法。另一种方法，第二个证明是一个建设性的正式推导，其中我们表示的平均不是一个积分，但作为一个pseudodiomatic运营商通过高斯矩母函数定义的行动。在扩展的斯泰因引理中，出现了概率学家的厄米多项式系数的绝对值，揭示了厄米多项式和正态分布之间的另一个联系。
英文摘要：
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.
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