SOURCE-LINKED INTELLIGENCE

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

arXiv · AI, language, vision and robotics · article · Sep 14, 2026 · UTC

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-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 force

Read original source ↗ Open in workspace

recordType

paper

region

Global