Isserlis-Wick theorem

Theorem Let $g$ be a gaussian random variable with variance ${\nu }^2$, $g\sim N(0,{\nu }^2)$. Let $F:\mathbb{R}\to \mathbb{R}$ be a differentiable function, and

$$\underset{x\to \pm \infty }{\lim }F(x)P(x)=0$$

where $P(x)$ is the density or $g$. Then,

$$\mathbb{E}[gF(g)]=\mathbb{E}(g^2)\mathbb{E}[F^{\prime }(g)]={\nu }^2\mathbb{E}[F^{\prime }(g)]$$

Proof Use integration by parts. $\square$

Remark It can be generalized to gaussian vectors:

Theorem Let $g$ be gaussian vectors and $F:{\mathbb{R}}^n\to \mathbb{R}$ a differentiables function, then

$$\mathbb{E}[g_{1}F(g)]=\sum _{l\le n}\mathbb{E}[g_1{g}_l]\mathbb{E}\left[\frac{\partial F(g)}{\partial g_j}\right]$$