Stein's Lemma

$\implies$ If $N \sim N(0,1)$, then \[ \mathbb{E}(f'(N)) = \mathbb{E}(Nf(N)) \] implies \[ \int_\mathbb{R} f'(x) \frac{1}{\sqrt{2\pi}}dx = \int_\mathbb{R} x f(x) \frac{1}{\sqrt{2\pi}}dx \] which if we translate into the Gaussian measure \[ \int_\mathbb{R} f'(x) \gamma(dx) = \int_\mathbb{R} xf(x) \gamma(dx) \] which is just a representation of integration by parts. Other direction is HW : TODO For some distributions, they are characterized by their moments. Necessarily this means the distribution has to be characterized by this. Distribution needs moments of all moments. If you figure it out what the moments, we can take $f$ to be a monomial then we get moments of higher order, since $\mathcal{H}$ is a seperating class.