MATH 7770: Stochastic Processes: Topics in Large Deviation Theory (fall 2008)

Instructor: John Mayberry

Many classical results in probability theory concern typical asymptotic behavior of stochastic processes. For instance, laws of large numbers identify appropriate limits for averages and central limit theorems deal with normal deviations from these limits. Large deviation theory takes a different approach by studying the limiting behavior of rare events. Early results related to such events date back to the 1930s with the work of Cramer in actuarial science (a sudden large claim leads to a payoff that an insurance company may not be ready to handle) and similar concepts have since found applications in a number of different fields including statistical mechanics, chemistry, hypothesis testing, financial mathematics, risk management, information theory, and neuroscience.

In this course, we will provide an introduction to the theory of large deviation principles starting with results for sequences of iid random variables (including Cramer's Theorem for sums, Sanov's Theorem for empirical measures, and specific examples). We will then develop techniques for large deviation principles in more general settings (main definitions, Varadhan's Lemma, tilting, and the contraction principle) and discuss large deviation results for Markov processes, general sequences of dependent random variables (Gärtner-Ellis Theorem), and Brownian motion with small diffusion coefficient (including the study of exit time problems and perturbations of dynamical systems). The remainder of the course will be dedicated to applications depending on the interests of students and time available.

This course is designed for students with a basic working knowledge of measure-based probability theory. Completion of a first-year course in graduate probability is sufficient and highly recommended background.