Polynomial Approximations of Conditional Expectations in Scalar Gaussian Channels

Abstract

We consider a channel $Y=X+N$ where $X$ is a random variable satisfying $\mathbb{E}[|X|]<\infty$ 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 $\mathbb{E}[X\mid Y],$ 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 $y\mapsto \mathbb{E}[X\mid Y=y]$ are expressible as multivariate polynomials in the functions $y\mapsto \mathbb{E}\left[{\left(X-\mathbb{E}[X\mid Y]\right)}^k\mid Y=y\right]$ for $k\in \mathbb{N}.$ 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 $\mathbb{E}[X\mid Y]$ by polynomials in $Y$ of degree at most $n$ decays faster than any polynomial in $n.$

A short version of this paper has been submitted to the 2021 IEEE International Symposium on Information Theory (ISIT)