Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity
Abstract
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 𝔼eβX2<∞ for some β>0, the scalar minimum mean-square error mmseX(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)=𝔼ezX. 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=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/2erm√(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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BibTeX
@misc{zhang2026zerosnr,
title = {{Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity}},
author = {Yixing Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15048},
eprint = {2609.15048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15048},
}
RIS
TY - UNPB
TI - Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity
AU - Yixing Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15048
UR - https://arxiv.org/abs/2609.15048
AB - Let Y_s=√sX+Z, where Z is standard Gaussian and independent of the real random variable X. We prove that, under the square-exponential moment condition 𝔼e^βX^2<∞ for some β>0, the scalar minimum mean-square error 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 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≥ 2 splits according to the roots of a Hermite polynomial and instead contributes a prefactor n^−m/2e^r_m√(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.
ER -