Stein’s lemma for a scalar variable is a well-known characterization of the Gaussian: x ∼ 𝒩(m,σ²) ⇔ ∀f 𝔼[(x−m)f(x)] = σ² 𝔼[f'(x)] Interestingly, there’s also a multivariate version, but it’s 2nd order: x ∼ 𝒩(m, σ² I) ⇔ ∀f 𝔼[(x−m)ᵀ∇f(x)] = σ² 𝔼[∇² f(x)] 1/2