Properties of zero-mean Gaussians

The following property of the Gaussian probability density functions (pdfs) is often used in this paper, here we state it in a form of a theorem:

Theorem 1   Let x $\in$ $\mathbb {R}$^m and p(x) zero-mean Gaussian pdf with covariance $\Sigma$ = {$\Sigma_{ij}^{}$} (i, j from 1 to m). If g : $\mathbb {R}$^m $\rightarrow$ $\mathbb {R}$ is a differentiable function not growing faster than a polynomial and with partial derivatives

$$\partial_{j}^{}$$g(x) = $${\frac{\partial}{\partial x_j}}$$g(x) ,

then

$$\int_{{\mathbb R}^m}^{} \sum_{j=1}^{m} \Sigma_{ij}^{} \int_{{\mathbb R}^m}^{} \partial_{j}^{}$$ (185)

In the following we will assume definite integration over $\mathbb {R}$^m whenever the integral appears. Alternatively, using the vector notation, the above identity reads:

$$\int \Sigma \int \nabla$$ (186)

For a general Gaussian pdf with mean $\mu$ the above equation transforms to:

$$\int \mu \int \Sigma \int \nabla$$ (187)

Proof. The proof uses the partial integration rule:

$$\int \nabla \int \nabla$$

where we have used the fast decay of the Gaussian function to dismiss one of the terms. Using the derivative of a Gaussian pdf. we have:

$$\int \nabla \int \Sigma^{-1}_{}$$

Multiplying both sides with $\Sigma$ leads to eq. (186), completing the proof. For the nonzero mean the deductions are also straightforward. $\qedsymbol$