Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity

Let $Y_s=\sqrt{s}X+Z$, where $Z$ is standard Gaussian and independent of the real random variable $X$. We prove that, under the square-exponential moment condition $\mathbb{E}e^{βX^2}0$, the scalar minimum mean-square error $\operatorname{mmse}_X(s)$ is analytic at zero signal-t