{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15048",
  "author_total": 1,
  "authors": [
    {
      "name": "Yixing Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zvbqvhxdw9ra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15048",
  "fulltext_url": null,
  "id": "7zvbqvhxdw9ra",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15048",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7213313815",
  "pdf_url": "https://arxiv.org/pdf/2609.15048",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zvbqvhxdw9ra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15048",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15048"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2613|Random Matrices and Applications",
      "subfield": "statistics"
    }
  ],
  "title": "Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity",
  "title_text": "Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7zvbqvhxdw9ra",
  "venue": "arXiv",
  "versions": []
}