Paper Links Zero-SNR Analyticity of Scalar MMSE to Gaussian Inputs
The paper studies a scalar channel where a real random variable X is observed under additive standard Gaussian noise scaled by the square root of a signal-to-noise parameter. Assuming a square-exponential moment condition on X, the author shows that the minimum mean-square error behaves analytically near zero SNR only when X is Gaussian, making that property equivalent to Gaussianity of the input. The result gives a way to detect non-Gaussian structure from the low-SNR behavior of the MMSE curve.