2021/02/11 by Wael Alghamdi, Alghamdi, Wael, Flávio P. Calmon +1 · 1 citation
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Probability (math.PR) #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2102.05970
openalex publication_date 2021/02/11 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider a channel Y=X+N where X is a random variable satisfying\n\𝔼[|X|]<\∞ and N is an independent standard normal random\nvariable. We show that the minimum mean-square error estimator of X from Y,\nwhich is given by the conditional expectation \𝔼[X \| Y], is a\npolynomial in Y if and only if it is linear or constant; these two cases\ncorrespond to X being Gaussian or a constant, respectively. We also prove\nthat the higher-order derivatives of y \↦ \𝔼[X \| Y=y] are\nexpressible as multivariate polynomials in the functions y \↦\n\𝔼\[ \( X - \𝔼[X \| Y] \)k \| Y = y \]\nfor k\∈ \ℕ. These expressions yield bounds on the 2-norm of the\nderivatives of the conditional expectation. These bounds imply that, if X has\na compactly-supported density that is even and decreasing on the positive\nhalf-line, then the error in approximating the conditional expectation\n\𝔼[X \| Y] by polynomials in Y of degree at most n decays\nfaster than any polynomial in n.\n