A general version of Price's theorem

TL;DR

This paper generalizes Price's theorem by proving the smoothness of the expectation of a nonlinear function of a Gaussian vector with arbitrary dimension and distributional nonlinearities, providing explicit derivative formulas and a rigorous proof.

Contribution

It unifies and rigorously proves Price's theorem for Gaussian vectors of any dimension with general nonlinearities modeled as tempered distributions.

Findings

Derived explicit formulas for derivatives of expectations with respect to covariance.

Established smoothness of the expectation map in the covariance matrix.

Extended Price's theorem to distributional nonlinear functions.

Abstract

Assume that $X_{\Sigma} \in \mathbb{R}^{n}$ is a centered random vector following a multivariate normal distribution with positive definite covariance matrix $\Sigma$. Let $g : \mathbb{R}^{n} \to \mathbb{C}$ be measurable and of moderate growth, say $|g(x)| \lesssim (1 + |x|)^{N}$. We show that the map $\Sigma \mapsto \mathbb{E}[g(X_{\Sigma})]$ is smooth, and we derive convenient expressions for its partial derivatives, in terms of certain expectations $\mathbb{E}[(\partial^{\alpha}g)(X_{\Sigma})]$ of partial (distributional) derivatives of $g$. As we discuss, this result can be used to derive bounds for the expectation $\mathbb{E}[g(X_{\Sigma})]$ of a nonlinear function $g(X_{\Sigma})$ of a Gaussian random vector $X_{\Sigma}$ 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 $n = 2$, it was generalized in 1964 by McMahon to the general case $g : \mathbb{R}^{2} \to \mathbb{C}$. Our contribution is to unify these results, and to give a mathematically fully rigorous proof. Precisely, we consider a normally distributed random vector $X_{\Sigma} \in \mathbb{R}^{n}$ 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 $\mathbb{E}\left[g(X_{\Sigma})\right]$ by the dual pairing $\left\langle g,\,\phi_{\Sigma}\right\rangle_{\mathcal{S}',\mathcal{S}}$, where $\phi_{\Sigma}$ denotes the probability density function of $X_{\Sigma}$.