A general version of Price's theorem

Assume that $X_{\mathrm{\Sigma }}\in {\mathbb{R}}^n$ is a centered random vector following a multivariate normal distribution with positive definite covariance matrix $\mathrm{\Sigma }$. Let $g:{\mathbb{R}}^n\to \mathbb{C}$ be measurable and of moderate growth, say $\mathrm{\mid }g(x)\mathrm{\mid }\lesssim (1+\mathrm{\mid }x\mathrm{\mid }{)}^N$. We show that the map $\mathrm{\Sigma }\mapsto \mathbb{E}[g(X_{\mathrm{\Sigma }})]$ is smooth, and we derive convenient expressions for its partial derivatives, in terms of certain expectations $\mathbb{E}[({\mathrm{\partial }}^{\alpha }g)(X_{\mathrm{\Sigma }})]$ of partial (distributional) derivatives of $g$. As we discuss, this result can be used to derive bounds for the expectation $\mathbb{E}[g(X_{\mathrm{\Sigma }})]$ of a nonlinear function $g(X_{\mathrm{\Sigma }})$ of a Gaussian random vector $X_{\mathrm{\Sigma }}$ with possibly correlated entries. For the case when $g\left(x\right)=g_1(x_1)\cdots g_n(x_n)$ has tensor-product structure, the above result is known in the engineering literature as Price’s theorem, originally published in 1958. For dimension $n = 2$,itwasgeneralizedin1964byMcMahontothegeneralcase$g : $\mathbb{R}$^{2} \to $\mathbb{C}$$. Our contribution is to unify these results, and to give a mathematically fully rigorous proof. Precisely, we consider a normally distributed random vector $X_{\mathrm{\Sigma }}\in {\mathbb{R}}^n$ of arbitrary dimension $n\in \mathbb{N}$, and we allow the nonlinearity $g$ to be a general tempered distribution. To this end, we replace the expectation $\mathbb{E}[g(X_{\mathrm{\Sigma }})]$ by the dual pairing $\left\langle g,,\phi_{\Sigma}\right\rangle_{$\mathcal{S}$',$\mathcal{S}$}$,where$\phi_{\Sigma}$denotestheprobabilitydensityfunctionof$X_{\Sigma}$.