A General Version of Price’s Theorem

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 .