## Thematic Program Probability and Its Applications
Fall 1998

## Graduate Course in Large Deviation Theory: Introduction and Applications

Instructor: Frank den Hollander denholla@sci.kun.nl

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.