This course is an introduction to large deviation theory, which is the part of probability theory that deals with the description of rare events. Examples are events where a sum of i.i.d. random variables deviates from its mean by more than a “normal amount”, i.e., beyond what is described by the central limit theorem.
A precise calculation of the probabilities of rare events turns out to be crucial for understanding a variety of different phenomena coming from physics, chemistry and biology.
The course consists of two parts: (A) Theory, (B) Applications. Part A begins with large deviations for sequences of i.i.d. random variables, then formulates the general framework of large deviation theory, and finally looks at large deviations for Markov sequences and Gibbs sequences. Part B describes applications in hypothesis testing, populations genetics, random networks and polymer chains.
The book from which the course is taught contains many exercises.
F. den Hollander, Large Deviations, Fields Institute Monographs 14, American Mathematical Society, Providence RI, 2000, x + 143 pp., ISBN 0-8218-1989-5