After successful completion of the course, students are able to explain the basic concepts and proof techniques in the theory of large deviations, of extreme values and of heavy-tailed random variables

This lecture is meant for bachelor students who are interested in probability theory and want to learn some advanced topics that are not treated in the Measure and Probability 1 and 2 lectures, or for master students. We will present several advanced topics chosen among the following:

• The theory of large deviations (that concerns the estimation 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. This part will include Cramer's and Sanov theorem, as well as applications to statistical physics: ferromagnetic behaviour in the Curie-Weiss model
• Extreme value theory and the classical laws: Gumbel, Weibull, Frechet
• Limit theorems for the sum of heavy-tailed random variables

The actual choice of topics can be adapted also according to the interests of the attending students

Lectures.

If less than 4 students (bachelor or master students) attend, 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

 Measure and probability theory I-II