Invariance property: Higher-order extension and application to Gaussian random variables

Ever since its introduction by Bussgang, the invariance property of Gaussian random variables has found numerous applications, for example, in statistical signal processing and digital communications. This property was later generalized using two different deterministic models for the zero-memory nonlinearity (ZMN): a Taylor series model with constant and structured (i.e. ZMN-based) coefficients and a general power series model with constant but totally unstructured coefficients. In this paper, we adopt a random power series model for the ZMN that generalizes all previous ones and then use it to address the higher-order extension of the invariance property of both non-Gaussian and Gaussian random variables. With this general model, the invariance is then shown to apply to both deterministic and random ZMNs. Moreover, our introduction of the helpful concept of a modified ZMN allows us to avoid using multi-dimensional techniques by casting our higher-order extension work in the same framework as that used for the derivations of the original first-order invariance property. A further contribution of this paper is made in the realm of Gaussian random variables where a new statistical equivalence between the ZMN and an MSE-optimal linear mapping is established, hence revealing that the Gaussianity of the two random variables used ensures an exact statistical linearization of the ZMN.