After successful completion of the course, students are able to explain the basic concepts and proof techniques in the theory of large deviations, and to understand its implications in a wide variety of applications.

The theory of large deviations concerns the fast (exponential) decay of the probability of certain "rare" or "extreme" events. The most typical example of such rare events occurs when the sample mean of a sequence of independent and identically distributed random variables deviates from its mean, even when the sample size grows large.

In the first part of the course, we will analyse the most important results and concepts in large deviations theory:

• Cramér's theorem
• general large deviation principles
• Sanov's theorem
• Gärtner-Ellis theorem

In the second part of the course we will deal with (some of) the following applications of large deviations theory:

• statistics: asymptotic optimality of Bayesian decision tests
• information theory: compression algorithms
• statistical physics: ferromagnetic behaviour in the Curie-Weiss model
• financial risk theory: ruin probability of an insurance company

Lectures.

If less than 4 students show up, the course will be cancelled and reannounced next year.

Because of the Covid pandemic, it is possible that the teaching modality will be changed to "online".

Oral exam

Books:

• den Hollander - Large Deviations
• Dembo, Zeitouni - Large Deviations Techniques and Applications

 Measure and probability theory I-II (but any students who know the basics of probability theory will be able to follow).