Non-Linear Transformations of Gaussians and Gaussian-Mixtures with implications on Estimation and Information Theory

This paper presents a useful theorem for non-linear transformations of the sum of independent, zeromean, Gaussian random variables. It is proved that the linear regression coefficient of the non-linear transformation output with respect to the overall input is identical to the linear regression coefficient with respect to any Gaussian random variable that is part of the input. As a side-result, the theorem is useful to simplify the computation of the partial regression coefficient also for non-linear transformations of Gaussian-mixtures. Due to its generality, and the wide use of Gaussians, and Gaussian-mixtures, to statistically model several phenomena, the potential use of the theorem spans multiple disciplines and applications, including communication systems, as well as estimation and information theory. In this view, the paper highlights how the theorem can be exploited to facilitate the derivation of fundamentals performance limits such as the SNR, the MSE and the mutual information in additive non-Gaussian (possibly non-linear) channels. Index Terms Gaussian random variables, Gaussian-mixtures, non-linearity, linear regression, SNR, MSE, mutual information.