Why Gaussianity?

Abstract

In this article, we try to answer the question: "Why the ubiquitous use and success of the Gaussian distribution law?". The history of the Gaussian or normal distribution is rather long, having existed for nearly 300 years since it was discovered by de Moivre in 1733, and the related literature is immense. An extended and thorough treatment of the topic and a survey of the works in the related area are given in the posthumously edited book of E.T. Jaynes (2003), and we partially follow this source, in particular while considering the history of the posed question. The important aspects of the general history of noise, especially of Brownian motion, are given by Cohen (2005). Our main contribution to the topic is concerned with highlighting the role of Gaussian models in signal processing based on the optimal property of the Gaussian distribution minimizing Fisher information over the class of distributions with a bounded variance. We deal only with the univariate Gaussian distribution, omitting the properties of multivariate Gaussian distribution. First of all, we present the ideas of classical derivations of the Gaussian law. Then we consider its properties and characterizations including the central limit theorem (CLT) and minimization of the distribution entropy and Fisher information. Finally, we dwell on the connections between Gaussianity and robustness in signal processing.