Gaussian Processes

This chapter explores the fundamental properties of Gaussian random variables and vectors and their distributions. It presents a comprehensive study of Gaussian processes, defining them and discussing the conditions for their existence. The chapter defines the Wiener process as a particular case of a Gaussian process, establishes its existence and recalls its fundamental properties. Building upon the concept of Gaussian variables, the chapter also introduces a distribution fundamental to statistical inference that arises directly from sums of squared Gaussian variables: the chi-squared distribution. When dealing with expectations of products of Gaussian random variables, especially for orders higher than two, a powerful tool known as Isserlis' theorem significantly simplifies calculations. With the general concepts of stochastic processes in place, the chapter precisely defines Gaussian processes, which are characterized by the Gaussian nature of all their finite-dimensional distributions.