Polynomial Approximations of Conditional Expectations in Scalar Gaussian Channels
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We consider a channel Y=X+N where X is a random variable satisfying 𝔼[|X|]<∞ and N is an independent standard normal random variable. We show that the minimum mean-square error estimator of X from Y, which is given by the conditional expectation 𝔼[X | Y], is a polynomial in Y if and only if it is linear or constant; these two cases correspond to X being Gaussian or a constant, respectively. We also prove that the higher-order derivatives of y ↦𝔼[X | Y=y] are expressible as multivariate polynomials in the functions y ↦𝔼[ ( X - 𝔼[X | Y] )^k | Y = y ] for k∈ℕ. These expressions yield bounds on the 2-norm of the derivatives of the conditional expectation. These bounds imply that, if X has a compactly-supported density that is even and decreasing on the positive half-line, then the error in approximating the conditional expectation 𝔼[X | Y] by polynomials in Y of degree at most n decays faster than any polynomial in n.
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