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- W2761127819 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 $gleft(xright) = g_{1}(x_{1}) cdots g_{n}(x_{n})$ 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 $g : mathbb{R}^{2} to mathbb{C}$. 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 $leftlangle g,,phi_{Sigma}rightrangle_{mathcal{S}',mathcal{S}}$, where $phi_{Sigma}$ denotes the probability density function of $X_{Sigma}$." @default.
- W2761127819 created "2017-10-20" @default.
- W2761127819 creator A5032519700 @default.
- W2761127819 date "2017-10-10" @default.
- W2761127819 modified "2023-10-01" @default.
- W2761127819 title "A general version of Price's theorem" @default.
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