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Stein's lemma relates Gaussian multiplication to differentiation. Let $Z\sim N(0,1)$. One sufficient set of conditions is that $g$ is locally absolutely continuous, both $\mathbb{E}|Zg(Z)|$ and $\mathbb{E}|g^{\prime }(Z)|$ are finite, and $g(x)\varphi (x)\to 0$ as $x\to \pm \infty$, where $\varphi$ is the standard normal density. ThenSince ${\varphi }^{\prime }(x)=-x\varphi (x)$, integration by parts on a finite interval moves the derivative from $\varphi$ to $g$; the boundary condition and integrability justify the infinite-interval limit.

For $g(z)=z$, the identity gives $\mathbb{E}{Z}^2=1$. For $g(z)=z^3$, it gives $\mathbb{E}{Z}^4=3\mathbb{E}{Z}^2=3$. More generally, for $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, standardization yieldsunder the corresponding conditions. The factor ${\sigma }^2$ is essential.

This integration-by-parts rule converts Gaussian error cross terms into derivatives of an estimator, which is the mechanism of the Stein risk identity for normal mean estimators. Boundary behavior matters for singular estimators: formal differentiation alone does not justify applying the identity across a nonintegrable singularity.