250189 VO Advanced probability theory (2021S)

The course starts by a definition of a probability space and spans crucial aspects of Probability Theory, including the expectation and the variance of a random variable, the law of large numbers and the central limit theorem, martingales and stopping times, finishing at Doob's martingale convergence theorem. The course also covers the required parts of Measure Theory.

The main topics:
- random variables, expectation, independence
- Borel-Cantelli lemmas, Kolmogorov zero-one law
- law of large numbers
- weak convergence
- central limit theorem
- martingales

An additional chapter of the course (not required for the exam) is an introduction to the percolation theory (https://en.wikipedia.org/wiki/Percolation_theory):
- classical theorems about the phase transition
- brief overview of recent progress (two Fields Medals, multiple breakthroughs)
- open questions
The aim of this chapter is to give a beautiful example of a probability space and to familiarize the audience with this exciting area of Mathematics.