Characteristic Functions of the Product of Two Gaussian Random Variables and the Product of a Gaussian and a Gamma Random Variable

We derive the characteristic function (CF) for two product distributions—first for the product of two Gaussian random variables (RVs), where one has zero mean and unity variance, and the other has arbitrary mean and variance. Next, we develop the characteristic function for the product of a gamma RV and a zero mean, unity variance Gaussian RV. The underlying rationale for this is to develop a model for a “quasi-Gaussian” RV—an RV that is nominally Gaussian, but with mean and variance parameters that are not constant, but instead, are RVs themselves. Due to the central limit theorem, many “real-world” processes are modeled as being Gaussian distributed. However, this implicitly assumes that the processes being modeled are perfectly stationary, which is often a poor assumption. The quasi-Gaussian model could be used as a more conservative description of many of these processes.