Assume that $$X_{\Sigma } \in \mathbb {R}^{n}$$ X Σ ∈ R n is a centered random vector following a multivariate normal distribution with positive definite covariance matrix $$\Sigma $$ Σ . Let $$g : \mathbb {R}^{n} \rightarrow \mathbb {C}$$ g : R n → C be measurable and of moderate growth, say $$|g(x)| \lesssim (1 + |x|)^{N}$$ | g ( x ) | ≲ ( 1 + | x | ) N . We show that the map $$\Sigma \mapsto \mathbb {E}[g(X_{\Sigma })]$$ Σ ↦ E [ g ( X Σ ) ] is smooth, and we derive convenient expressions for its partial derivatives, in terms of certain expectations $$\mathbb {E}[(\partial ^{\alpha }g)(X_{\Sigma })]$$ E [ ( ∂ α g ) ( X Σ ) ] of partial (distributional) derivatives of g . As we discuss, this result can be used to derive bounds for the expectation $$\mathbb {E}[g(X_{\Sigma })]$$ E [ g ( X Σ ) ] of a nonlinear function $$g(X_{\Sigma })$$ g ( X Σ ) of a Gaussian random vector $$X_{\Sigma }$$ X Σ with possibly correlated entries. For the case when $$g\left( x\right) = g_{1}(x_{1}) \cdots g_{n}(x_{n})$$ g x = g 1 ( x 1 ) ⋯ g n ( x n ) has tensor-product structure, the above result is known in the engineering literature as Price’s theorem , originally published in 1958. For dimension $$n = 2$$ n = 2 , it was generalized in 1964 by McMahon to the general case $$g : \mathbb {R}^{2} \rightarrow \mathbb {C}$$ g : R 2 → C . Our contribution is to unify these results, and to give a mathematically fully rigorous proof. Precisely, we consider a normally distributed random vector $$X_{\Sigma } \in \mathbb {R}^{n}$$ X Σ ∈ R n of arbitrary dimension $$n \in \mathbb {N}$$ n ∈ N , and we allow the nonlinearity g to be a general tempered distribution. To this end, we replace the expectation $$\mathbb {E}\left[ g(X_{\Sigma })\right] $$ E g ( X Σ ) by the dual pairing $$\left\langle g,\,\phi _{\Sigma }\right\rangle _{\mathcal {S}',\mathcal {S}}$$ g , ϕ Σ S ′ , S , where $$\phi _{\Sigma }$$ ϕ Σ denotes the probability density function of $$X_{\Sigma }$$ X Σ .

中文翻译：

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普莱斯定理的一般版本

假设 $$X_{\Sigma } \in \mathbb {R}^{n}$$ X Σ ∈ R n 是一个以正定协方差矩阵 $$\Sigma $$ Σ 服从多元正态分布的中心随机向量。假设 $$g : \mathbb {R}^{n} \rightarrow \mathbb {C}$$ g : R n → C 是可测量的并且是适度增长的，比如 $$|g(x)| \lesssim (1 + |x|)^{N}$$ | 克 ( x ) | ≲ ( 1 + | x | ) N 。我们证明了映射 $$\Sigma \mapsto \mathbb {E}[g(X_{\Sigma })]$$ Σ ↦ E [ g ( X Σ ) ] 是平滑的，并且我们推导出其偏导数的方便表达式，在某些期望方面 $$\mathbb {E}[(\partial ^{\alpha }g)(X_{\Sigma })]$$ E [ (∂ α g ) ( X Σ ) ] 的部分（分布) g 的导数。在我们讨论时，该结果可用于推导出非线性函数 $$g(X_{\Sigma) 的期望值 $$\mathbb {E}[g(X_{\Sigma })]$$ E [ g ( X Σ ) ] 的边界})$$ g ( X Σ ) 的高斯随机向量 $$X_{\Sigma }$$ X Σ 可能相关条目。对于 $$g\left( x\right) = g_{1}(x_{1}) \cdots g_{n}(x_{n})$$ gx = g 1 ( x 1 ) ⋯ gn ( xn ) 具有张量积结构，上述结果在工程文献中称为普莱斯定理，最初发表于 1958 年。对于维度 $$n = 2$$ n = 2 ，它在 1964 年由 McMahon 推广到一般情况$$g : \mathbb {R}^{2} \rightarrow \mathbb {C}$$ g : R 2 → C 。我们的贡献是统一这些结果，并给出数学上完全严格的证明。恰恰，我们考虑一个正态分布的随机向量 $$X_{\Sigma } \in \mathbb {R}^{n}$X Σ ∈ R n 任意维度 $$n \in \mathbb {N}$$ n ∈ N ，我们允许非线性 g 是一个一般的调和分布。为此，我们将期望值 $$\mathbb {E}\left[ g(X_{\Sigma })\right] $$ E g ( X Σ ) 替换为对偶对 $$\left\langle g,\ ,\phi _{\Sigma }\right\rangle _{\mathcal {S}',\mathcal {S}}$$ g , ϕ Σ S ′ , S , 其中 $$\phi _{\Sigma }$$ ϕ Σ 表示 $$X_{\Sigma }$$ X Σ 的概率密度函数。