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A General Version of Price’s Theorem

A Tool for Bounding the Expectation of Nonlinear Functions of Gaussian Random Vectors

Voigtlaender, Felix

Assume that XΣ∈Rn is a centered random vector following a multivariate normal distribution with positive definite covariance matrix Σ. Let g:Rn→C be measurable and of moderate growth, say |g(x)|≲(1+|x|)N. We show that the map Σ↦E[g(XΣ)] is smooth, and we derive convenient expressions for its partial derivatives, in terms of certain expectations E[(∂αg)(XΣ)] of partial (distributional) derivatives of g. As we discuss, this result can be used to derive bounds for the expectation E[g(XΣ)] of a nonlinear function g(XΣ) of a Gaussian random vector XΣ with possibly correlated entries. For the case when g(x)=g1(x1)⋯gn(xn) 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:R2→C. Our contribution is to unify these results, and to give a mathematically fully rigorous proof. Precisely, we consider a normally distributed random vector XΣ∈Rn of arbitrary dimension n∈N, and we allow the nonlinearity g to be a general tempered distribution. To this end, we replace the expectation E[g(XΣ)] by the dual pairing ⟨g,ϕΣ⟩S′,S, where ϕΣ denotes the probability density function of XΣ.
Published in: Journal of Theoretical Probability, 10.1007/s10959-020-01017-w, Springer