We prove a formula for the evaluation of averages containing a scalar function of a Gaussian random vector multiplied by a product of the random vector components, each one raised at a power. Some powers could be of zeroth-order, and, for averages containing only one vector component to the first power, the formula reduces to Stein's lemma for the multivariate normal distribution. Also, by setting the said function inside average equal to one, we easily derive Isserlis theorem and its generalizations, regarding higher order moments of a Gaussian random vector. We provide two proofs of the formula, with the first being a rigorous proof via mathematical induction. The second is a formal, constructive derivation based on treating the average not as an integral, but as the action of pseudodifferential operators defined via the moment-generating function of the Gaussian random vector.

翻译：我们证明一个公式用于评估包含高斯随机矢量的星际函数的平均值。 该公式由高斯随机矢量的成份乘以随机矢量组件的产物, 每种矢量在动力中生成。 有些权力可能是零级的, 而对于仅包含第一个动力中一个矢量组件的平均值, 该公式在多变量正常分布中降为施泰因的列马值。 另外, 通过将上述函数设置在平均值中等于一个, 我们很容易得出Isserlis 定理及其一般化, 有关高斯随机矢量的更高顺序时刻。 我们提供了两种公式的证明, 第一项是数学感应的严格证明。 第二项是正式的、 建设性的推断, 其依据是将平均值不作为整体, 而是作为通过高斯随机矢量的瞬时函数定义的伪差异操作器的动作。