标量MMSE在零信噪比处的解析性等价于高斯性

中文摘要 AI 辅助

设$Y_s=\sqrt{s}X+Z$，其中$Z$为标准高斯分布且与实随机变量$X$独立。我们证明，在平方指数矩条件$\mathbb{E}e^{\beta X^2}<\infty$（对某个$\beta>0$）下，标量最小均方误差$\operatorname{mmse}_X(s)$在零信噪比处解析当且仅当$X$为高斯分布，其中常数随机变量作为退化高斯分布包含在内。证明将高斯信道中的估计问题转化为作用于矩生成函数$M(z)=\mathbb{E}e^{zX}$上的反向热流。在所给尾部条件下，每个非高斯输入迫使$M$具有非零复零点。我们证明每个零点簇在其局部Borel变换中于作用量$\xi=z_0^2/2$处产生有限奇异性。去除作用量尺度后，对于简单零点，Borel系数具有非零的$n^{-1/2}$前因子。重数$m\geq 2$的零点根据Hermite多项式的根分裂，并贡献前因子$n^{-m/2}e^{r_m\sqrt{2n}}$。有限圆盘局部化和相对环延拓论证表明，至少有一个这样的奇异性在完整Borel变换中存留。因此，对于所述类别中的每个非高斯输入，形式零信噪比展开是Gevrey-1但发散的。有理MMSE刚性和互信息的类似解析性判据作为推论得出。

英文摘要

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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33 pages, 1 figure