A General Version of Price\u2019s Theorem

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} rightarrow 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 gleft( xright) = 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} rightarrow 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 leftlangle g,,phi _{Sigma }rightrangle _{mathcal {S}',mathcal {S}}, where phi _{Sigma } denotes the probability density function of X_{Sigma }.