Another Useful Theorem for Non-Linear Transformations of Gaussian Random Variables

This paper presents a useful theorem for non-linear transformations of the sum of independent zero-mean Gaussian random variables. It is proved that the linear regression coefficient of the non-linear transformation output with respect to any Gaussian random variable that is part of its input, is identical to the linear regression coefficient with respect to the overall input. As a side-result, the theorem is particularly useful to simplify the computation of the partial regression coefficients not only when the input is Gaussian distributed, but also for non-linear transformations of Gaussian-Mixtures. Due to its generality, and the wide use of Gaussian and Gaussian-Mixture probability density functions to statistically model several physical phenomenon, the potential use of the theorem spans multiple disciplines and applications, including communication systems, as well as estimation and information theory.