July 20, 2024

Thematic Program Probability and Its Applications Fall 1998

Graduate Course in Large Deviation Theory: Introduction and Applications

Instructor: Frank den Hollander
Department of Mathematics, University of Nijmegen

starting Sept. 14, 1998

Prerequisite: Basic probability theory.

This course is an introduction to large deviations and describes both theory and applications. Large deviation theory -- a part of probability, statistics and statistical physics -- deals with the description of events where a random variable deviates from its mean more than a "normal" amount (ie. beyond what is described by the central limit theorem). A precise calculation of the probabilities of such events turns out to be crucial for the study of integrals of exponential functionals of random variables, which come up in many different contexts.

Part 1: Derivation of some elementary large deviation theorems for i.i.d random variables. Here the emphasis is on explicit calculation for several simple examples and on understanding of the basic mechanisims in force.

Part 2: Presentation of some general definitions and theorems in a
more abstract context. Here the goal is to expose a unified scheme that gives large deviation theory its general structure and that can be made to work in a variety of cases.

Part 3: Derivations of some elementary large deviation theorems for Markov chains, again with the emphasis on explicit calculation.

Part 4: Description of applications. Some examples include: polymer chains, statistical hypothesis testing and neural networks.

Evaluation: Many exercises available.