论文标题
多元正态分布的时刻的新公式扩展了Stein的引理和Isserlis定理
New formulas for moments of the multivariate normal distribution extending Stein's lemma and Isserlis theorem
论文作者
论文摘要
我们证明了一种评估包含高斯随机矢量标量函数的期望的公式，乘以随机矢量成分的产物，每个载体都提高到非负整数功率。某些功率可能是零秩序的，并且对于仅包含一个向量组件的期望，公式将其减少到Stein的引理中，以使多变量正态分布。此外，通过将功能设置为期望等于1的功能，我们就可以轻松地对高斯随机向量的高阶矩进行重新启用Isserlis定理及其概括。我们提供了两个公式的证据，第一个是通过数学诱导的严格证明。第二个证明是基于将期望视为积分而而是作为正式的建设性推导，而是作为通过高斯随机向量的矩造成函数定义的假差异操作员的连续动作。
We prove a formula for the evaluation of expectations containing a scalar function of a Gaussian random vector multiplied by a product of the random vector components, each one raised to a non-negative integer power. Some of the powers could be of zeroth order, and, for expectations containing only one vector component to the first power, the formula reduces to Stein's lemma for the multivariate normal distribution. Furthermore, by setting the function inside expectation equal to one, we easily re-derive Isserlis theorem and its generalizations, regarding higher-order moments of a Gaussian random vector. We provide two proofs of the formula, the first being a rigorous proof via mathematical induction. The second proof is a formal, constructive derivation based on treating the expectation not as an integral, but as the consecutive actions of pseudodifferential operators defined via the moment-generating function of the Gaussian random vector.
