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

Let Ys=√sX+Z, where Z is standard Gaussian and independent of the real random variable X. We prove that, under the square-exponential moment condition mathbbEeβ X2<∞ for some β>0, the scalar minimum mean-square error operatornamemmseX(s) is analytic at zero signal-to-noise ratio if and only if X is Gaussian, with constant random variables included as degenerate Gaussians. The proof converts estimation in the Gaussian channel into a backward heat flow acting on the moment-generating function M(z)=mathbbEezX. Under the stated tail condition, every non-Gaussian input forces M to have a nonzero complex zero. We show that each zero cluster produces a finite singularity in its localized Borel transform at the action xi=z02/2. After removing the action scale, the Borel coefficients have a nonzero n-1/2 prefactor for a simple zero. A zero of multiplicity m≥ 2 splits according to the roots of a Hermite polynomial and instead contributes a prefactor n-m/2e^rm√2n. A finite-disc localization and relative-cycle continuation argument then show that at least one such singularity survives in the full Borel transform. Thus, for every non-Gaussian input in the stated class, the formal zero-SNR expansion is Gevrey-1 but divergent. Rational-MMSE rigidity and the analogous analyticity criterion for mutual information follow as corollaries.