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

arXiv:2609.15048v1 Announce Type: cross Abstract: 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 Ee^{eta X^2}0, the scalar minimum mean-square error operatorname{mmse}_X(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)=Ee^{zX}. 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=z_0^2/2. After removing the action scale, the Borel coefficients have a nonzero n^{-1/2} prefactor for a simple zero. A zero of multiplicity mgeq 2 splits according to the roots of a Hermite polynomial and instead contributes a prefactor n^{-m/2}e^{r_msqrt{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.

Source: arXiv cs.LG | 2026-09-15