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

• URL: http://arxiv.org/abs/2609.15048v1
• Date: Mon, 14 Sep 2026 05:07:20 GMT
• Title: Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity
• Abstract summary: We prove that the minimum mean-square error $operatornamemmse_X(s)$ is analytic at zero signal-to-noise ratio if and only if $X$ is Gaussian.<n>A zero of $mgeq 2$ splits according to the roots of a Hermitemath and instead contributes a prefactor $n-m/2er_msqrt2n$.
• Score: 0.7499255435598924
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\mathbb{E}e^{βX^2}<\infty$ for some $β>0$, 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 $M(z)=\mathbb{E}e^{zX}$. 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 $ξ=z_0^2/2$. After removing the action scale, the Borel coefficients have a nonzero $n^{-1/2}$ prefactor for a simple zero. A zero of multiplicity $m\geq 2$ splits according to the roots of a Hermite polynomial and instead contributes a prefactor $n^{-m/2}e^{r_m\sqrt{2n}}$. 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.

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