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This paper investigates the statistical properties of non-lineartransformations (NLT) of random variables, in order to establish useful toolsfor estimation and information theory. Specifically, the paper focuses onlinear regression analysis of the NLT output and derives sufficient generalconditions to establish when the input-output regression coefficient is equalto the \emph{partial} regression coefficient of the output with respect to a(additive) part of the input. A special case is represented by zero-meanGaussian inputs, obtained as the sum of other zero-mean Gaussian randomvariables. The paper shows how this property can be generalized to theregression coefficient of non-linear transformations of Gaussian-mixtures. Dueto its generality, and the wide use of Gaussians and Gaussian-mixtures tostatistically model several phenomena, this theoretical framework can findapplications in multiple disciplines, such as communication, estimation, andinformation theory, when part of the nonlinear transformation input is thequantity of interest and the other part is the noise. In particular, the papershows how the said properties can be exploited to simplify closed-formcomputation of the signal-to-noise ratio (SNR), the estimation mean-squarederror (MSE), and bounds on the mutual information in additive non-Gaussian(possibly non-linear) channels, also establishing relationships among them.

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