Analysis Seminar

Title: Introduction to Gaussian Processes
Speaker: Robbie Robinson, George Washington University
Place: Monroe 267

Abstract: Stationary Gaussian processes are classical probabilistic models used in applications such as digital signal processing, machine learning and quantum mechanics. In the 1950s Girsanov introduced stationary Gaussian processes into ergodic theory as a class of examples with spectral properties that could, to some extent, be prescribed. Since then, Gaussian processes seem to have resurfaced in ergodic theory about once a decade to solve new problems. Yet from the ergodic theory point of view, Gaussian processes remain a somewhat peculiar class, still somewhat poorly understood, and largely distinct from the typical examples.

In this first talk, in what will be an intermittent set of talks, I will set up the measure theoretic background for studying probability theory and stochastic processes in general, and in the context of ergodic theory, including a discussion the Kolmogorov consistency theorem. I will also describe the special properties of Gaussian distributions that make Gaussian processes a nice kind of stochastic process to study. Later talks will discuss topics like Gaussian Hilbert spaces, spectral theory via Fock spaces and Weiner chaos, and the problems associated with moving from discrete to continuous time, and from real to complex valued random variables.