**Prerequisites**

The 'standard' probability and analysis courses taught in the mathematics BSc are prerequisites for this course. It is strongly advised to have taken another course with probabilistic content e.g. on martingales, advanced probability or Markov chains. Measure theory and Lebesgue integration theory is built up 'from scratch' in the first five weeks of this course. However, the course is quite tough for those students who have not done any measure- and integration theory previously. In that case it may help to consult RenĂ© Schilling's Measures, Integrals and Martingales, or Jeffrey Rosenthal's A First Look at Rigorous Probability Theory for some extra material on this topic. We will use lectures notes of Prof. Peter Spreij.

**Aim of the course**

- Discuss the measure- and Lebesgue integration theory that is relevant in probability theory.
- Introduce some vital concepts in probability theory, such as conditional expectations, the Radon-Nikodym theorem, martingale convergence theorems, characteristic functions and why they are characteristic, construction of Brownian motion (if time permits).
- Provide rigorous proofs for two central convergence theorems in probability: the Strong Law of Large Numbers and the generalized Central Limit Theorem.

**Lecturers**

Dalia Theresiu (d.e.terhesiu@math.leidenuniv.nl) and Wioletta Ruszel (w.m.ruszel@uu.nl)

**Teaching Assistant**

TBA

- Docent: Jonny Imbierski
- Docent: Wioletta Ruszel
- Docent: Olov Schavemaker
- Docent: Dalia Terhesiu